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BEGIN:VEVENT
SUMMARY:Andrew Granville (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20200430T150000Z
DTEND;VALUE=DATE-TIME:20200430T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/1
DESCRIPTION:Title: Frobenius's postage stamp problem\, and beyond...\nby Andrew Gran
ville (Université de Montréal) as part of Number Theory Web Seminar\n\n\
nAbstract\nWe study this famous old problem from the modern perspective of
additive combinatorics\, and then look at generalizations.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20200507T150000Z
DTEND;VALUE=DATE-TIME:20200507T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/2
DESCRIPTION:Title: Sums of three cubes\nby Andrew Sutherland (MIT) as part of Number
Theory Web Seminar\n\n\nAbstract\nIn 1953 Mordell asked whether one can r
epresent 3 as a sum of three cubes in any way other than $1^3+1^3+1^3$ and
$4^3+4^3 -5^3$. Mordell's question spurred many computational investigati
ons over the years\, and while none found a new solution for 3\, they even
tually determined which of the first 100 positive integers $k$ can be repr
esented as a sum of three cubes in all but one case: $k=42$.\n\nIn this ta
lk I will present joint work with Andrew Booker that used Charity Engine's
crowd-sourced compute grid to affirmatively answer Mordell's question\, a
s well as settling the case $k=42$. I will also discuss a conjecture of He
ath-Brown that predicts the existence of infinitely many more solutions an
d explains why they are so difficult to find.\n\nMSC:11Y50\, MSC:11D25\, A
CM:F.2.2\, ACM:G.2.3\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Blomer (Universität Bonn)
DTSTART;VALUE=DATE-TIME:20200514T150000Z
DTEND;VALUE=DATE-TIME:20200514T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/3
DESCRIPTION:Title: Joint equidistribution and fractional moments of L-functions\nby
Valentin Blomer (Universität Bonn) as part of Number Theory Web Seminar\n
\n\nAbstract\nIntegral points on spheres of large radius $D^{1/2}$ equidst
ribute (subject to appropriate congruence conditions)\, and so do Heegner
points of large discriminant $D$ on the modular curve. Both sets have roug
hly the same cardinality\, and there is a natural way to associate with ea
ch point on the sphere a Heegner point. Do these pairs equidstribute in th
e product space of the sphere and the modular curve as $D$ tends to infini
ty?\n\nA seemingly very different\, but structurally similar joint equidis
tribution problem can be asked for the supersingular reduction at two diff
erent primes of elliptic curves with CM by an order of large discriminant
$D$.\n\nBoth equidistribution problems have been studied by ergodic method
s under certain conditions on $D$. I will explain how to use number theory
and families of high degree $L$-functions to obtain an effective equidist
ribution statement with a rate of convergence\, assuming GRH. This is join
t work in progress with F. Brumley.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Waldschmidt (Sorbonne University)
DTSTART;VALUE=DATE-TIME:20200512T080000Z
DTEND;VALUE=DATE-TIME:20200512T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/4
DESCRIPTION:Title: Representation of integers by cyclotomic binary forms\nby Michel
Waldschmidt (Sorbonne University) as part of Number Theory Web Seminar\n\n
\nAbstract\nThe representation of positive integers as a sum of two square
s is a classical problem studied by Landau and Ramanujan. A similar result
has been obtained by Bernays for positive definite binary form. In joint
works with Claude Levesque and Etienne Fouvry\, we consider the representa
tion of integers by the binary forms which are deduced from the cyclotomic
polynomials. One main tool is a recent result of Stewart and Xiao which g
eneralizes the theorem of Bernays to binary forms of higher degree.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Voloch (University of Canterbury)
DTSTART;VALUE=DATE-TIME:20200609T000000Z
DTEND;VALUE=DATE-TIME:20200609T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/6
DESCRIPTION:Title: Value sets of sparse polynomials\nby Felipe Voloch (University of
Canterbury) as part of Number Theory Web Seminar\n\n\nAbstract\nWe obtain
a lower bound on the size of the value set $f(F_p)$ of a sparse polynomia
l $f(x)$ in $F_p[x]$ over a finite field of $p$ elements when $p$ is prime
. This bound is uniform with respect to the degree and depends on the numb
er of terms of $f$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timothy Browning (IST Austria)
DTSTART;VALUE=DATE-TIME:20200604T150000Z
DTEND;VALUE=DATE-TIME:20200604T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/7
DESCRIPTION:Title: Random Diophantine equations\nby Timothy Browning (IST Austria) a
s part of Number Theory Web Seminar\n\n\nAbstract\nI’ll survey some of t
he key challenges around the solubility of polynomial Diophantine equation
s over the integers.\n\nWhile studying individual equations is often extra
ordinarily difficult\, the situation is more accessible if we merely ask w
hat happens on average and if we restrict to the so-called Fano range\, wh
ere the number of variables exceeds the degree of the polynomial. Indeed\
, about 20 years ago\, it was conjectured by Poonen and Voloch that random
Fano hypersurfaces satisfy the Hasse principle\, which is the simplest ne
cessary condition for solubility. After discussing related results I’ll
report on joint work with Pierre Le Boudec and Will Sawin where we establ
ish this conjecture for all Fano hypersurfaces\, except cubic surfaces.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (IAS and Princeton University)
DTSTART;VALUE=DATE-TIME:20200625T150000Z
DTEND;VALUE=DATE-TIME:20200625T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/8
DESCRIPTION:Title: Integer points on affine cubic surfaces\nby Peter Sarnak (IAS and
Princeton University) as part of Number Theory Web Seminar\n\n\nAbstract\
nThe level sets of a cubic polynomial in four or more variables tends to h
ave many integer solutions\, while ones in two variables a limited number
of solutions. Very little is known in case of three variables. For cubics
which are character varieties (thus carrying a nonlinear group of morphism
s) a Diophantine analysis has been developed and we will describe it. Pass
ing from solutions in integers to integers in say a real quadratic field t
here is a fundamental change which is closely connected to challenging que
stions about one-commutators in $SL_2$ over such rings.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kristin Lauter (Microsoft Research Redmond Labs)
DTSTART;VALUE=DATE-TIME:20200519T000000Z
DTEND;VALUE=DATE-TIME:20200519T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/9
DESCRIPTION:Title: How to keep your secrets in a post-quantum world\nby Kristin Laut
er (Microsoft Research Redmond Labs) as part of Number Theory Web Seminar\
n\n\nAbstract\nAs we move towards a world which includes quantum computers
which exist at scale\, we are forced to consider the question of what har
d problems in mathematics our next generation of cryptographic systems wil
l be based on. Supersingular Isogeny Graphs were proposed for use in cryp
tography in 2006 by Charles\, Goren\, and Lauter. Supersingular Isogeny G
raphs are examples of Ramanujan graphs\, which are optimal expander graphs
. These graphs have the property that relatively short walks on the grap
h approximate the uniform distribution\, and for this reason\, walks on ex
pander graphs are often used as a good source of randomness in computer sc
ience. But the reason these graphs are important for cryptography is that
finding paths in these graphs\, i.e. routing\, is hard: there are no know
n subexponential algorithms to solve this problem\, either classically or
on a quantum computer. For this reason\, cryptosystems based on the hardn
ess of problems on Supersingular Isogeny Graphs are currently under consid
eration for standardization in the NIST Post-Quantum Cryptography (PQC) Co
mpetition. This talk will introduce these graphs\, the cryptographic appl
ications\, and the various algorithmic approaches which have been tried to
attack these systems.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel-Aviv University)
DTSTART;VALUE=DATE-TIME:20200521T150000Z
DTEND;VALUE=DATE-TIME:20200521T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/10
DESCRIPTION:Title: Prime lattice points in ovals\nby Zeev Rudnick (Tel-Aviv Univers
ity) as part of Number Theory Web Seminar\n\n\nAbstract\nThe study of the
number of lattice points in dilated regions has a long history\, with seve
ral outstanding open problems. In this lecture\, I will describe a new var
iant of the problem\, in which we study the distribution of lattice points
with prime coordinates. We count lattice points in which both coordinates
are prime\, suitably weighted\, which lie in the dilate of a convex plana
r domain having smooth boundary\, with nowhere vanishing curvature. We obt
ain an asymptotic formula\, with the main term being the area of the dilat
ed domain\, and our goal is to study the remainder term. Assuming the Riem
ann Hypothesis\, we give a sharp upper bound\, and further assuming that t
he positive imaginary parts of the zeros of the Riemann zeta functions are
linearly independent over the rationals allows us to give a formula for t
he value distribution function of the properly normalized remainder term.
(joint work with Bingrong Huang).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART;VALUE=DATE-TIME:20200528T150000Z
DTEND;VALUE=DATE-TIME:20200528T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/11
DESCRIPTION:Title: Bracket quadratics\, Hua’s Lemma and Vinogradov’s mean value the
orem\nby Trevor Wooley (Purdue University) as part of Number Theory We
b Seminar\n\n\nAbstract\nA little over a decade ago\, Ben Green posed the
problem of showing that all large integers are the sum of at most a bounde
d number of bracket quadratic polynomials of the shape $n[n\\theta]$\, for
natural numbers $n$\, in which $\\theta$ is an irrational number such as
the square-root of 2. This was resolved in the PhD thesis of Vicky Neale\,
although no explicit bound was given concerning the number of variables r
equired to achieve success. In this talk we describe a version of Hua’s
lemma for this problem that can be applied via the Hardy-Littlewood method
to obtain a conclusion with 5 variables. The associated argument differs
according to whether $\\theta$ is a quadratic irrational or not. We also e
xplain how related versions of Hua’s lemma may be interpreted in terms o
f discrete restriction variants of Vinogradov’s mean value theorem\, thu
s providing a route to generalisation.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elon Lindenstrauss (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20200618T150000Z
DTEND;VALUE=DATE-TIME:20200618T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/12
DESCRIPTION:Title: Diagonalizable flows\, joinings\, and arithmetic applications\nb
y Elon Lindenstrauss (Hebrew University of Jerusalem) as part of Number Th
eory Web Seminar\n\n\nAbstract\nRigidity properties of higher rank diagona
lizable actions have proved to be powerful tools in understanding the dist
ribution properties of rational tori in arithmetic quotients. Perhaps the
simplest\, and best known\, example of such an equidistribution question i
s the equidistribution of CM points of a given discriminant on the modular
curve. The equidistribution of CM points was established by Duke using an
alytic methods\, but for finer questions (and questions regarding equidist
ribution on higher rank spaces) the ergodic theoretic approach has proved
to be quite powerful.\n\nI will survey some of the results in this directi
on\, including several results about joint distributions of collections of
points in product spaces by Aka\, Einsiedler\, Khayutin\, Shapira\, Wiese
r and other researchers.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (University of Oxford)
DTSTART;VALUE=DATE-TIME:20200702T150000Z
DTEND;VALUE=DATE-TIME:20200702T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/13
DESCRIPTION:Title: Primes in arithmetic progressions to large moduli\nby James Mayn
ard (University of Oxford) as part of Number Theory Web Seminar\n\n\nAbstr
act\nHow many primes are there which are less than $x$ and congruent to $a
$ modulo $q$? This is one of the most important questions in analytic numb
er theory\, but also one of the hardest - our current knowledge is limited
\, and any direct improvements require solving exceptionally difficult que
stions to do with exceptional zeros and the Generalized Riemann Hypothesis
!\n\nIf we ask for 'averaged' results then we can do better\, and powerful
work of Bombieri and Vinogradov gives good answers for $q$ less than the
square-root of $x$. For many applications this is as good as the Generaliz
ed Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a
notorious problem which has been achieved only in special situations\, pe
rhaps most notably this was the key component in the work of Zhang on boun
ded gaps between primes. I'll talk about recent work going beyond this bar
rier in some new situations. This relies on fun connections between algebr
aic geometry\, spectral theory of automorphic forms\, Fourier analysis and
classical prime number theory. The talk is intended for a general audienc
e.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW Sydney)
DTSTART;VALUE=DATE-TIME:20200623T080000Z
DTEND;VALUE=DATE-TIME:20200623T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/14
DESCRIPTION:Title: Weyl sums: large\, small and typical\nby Igor Shparlinski (UNSW
Sydney) as part of Number Theory Web Seminar\n\n\nAbstract\nAbstract: Whil
e Vinogradov’s Mean Value Theorem\, in the form given by J. Bourgain\, C
. Demeter and L. Guth (2016) and T. Wooley (2016-2019)\, gives an essentia
lly optimal result on the power moments of the Weyl sums \n$$\nS(u\;N) =\
\sum_{1\\le n \\le N} \\exp(2 \\pi i (u_1n+…+u_dn^d))\n$$\nwhere $u = (u
_1\,...\,u_d) \\in [0\,1)^d$\, very little is known about the distributio
n\, or even existence\, of $u \\in [0\,1)^d$\, for which these sums are ve
ry large\, or small\, or close to their average value $N^{1/2}$. In this t
alk\, we describe recent progress towards these and some related questions
.\n\nWe also present some new bounds on $S(u\;N)$ which hold for almost al
l $(u_i)_{i\\in I}$ and all $(u_j)_{j\\in J}$\, where $I \\cup J$ is a par
tition of $\\{1\,…\,\,d\\}$. These bounds improve similar results of T.
Wooley (2015). Our method also applies to binomial sums \n$$\nT(x\,y\; N)
= \\sum_{1\\le n \\le N} \\exp(2 \\pi i (xn+yn^d))\n$$\nwith $x\,y \\in [0
\,1)$\, in which case we improve some results of M.B. Erdogan and G. Shaka
n (2019).\n\nThis is a joint work with Changhao Chen and Bryce Kerr.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20200716T150000Z
DTEND;VALUE=DATE-TIME:20200716T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/15
DESCRIPTION:Title: A tale of three curves\nby Jennifer Balakrishnan (Boston Univers
ity) as part of Number Theory Web Seminar\n\n\nAbstract\nWe will describe
variants of the Chabauty-Coleman method\nand quadratic Chabauty to determi
ne rational points on curves. In so\ndoing\, we will highlight some recent
examples where the techniques\nhave been used: this includes a problem of
Diophantus originally\nsolved by Wetherell and the problem of the "cursed
curve"\, the split\nCartan modular curve of level 13. This is joint work
with Netan Dogra\,\nSteffen Mueller\, Jan Tuitman\, and Jan Vonk.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Bilu (University of Bordeaux)
DTSTART;VALUE=DATE-TIME:20200611T150000Z
DTEND;VALUE=DATE-TIME:20200611T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/16
DESCRIPTION:Title: Trinomials\, singular moduli and Riffaut's conjecture\nby Yuri B
ilu (University of Bordeaux) as part of Number Theory Web Seminar\n\n\nAbs
tract\nRiffaut (2019) conjectured that a singular modulus of degree h>2 ca
nnot be a root of a trinomial with rational coefficients. We show that thi
s conjecture follows from the GRH\, and obtain partial unconditional resul
ts. A joint work with Florian Luca and Amalia Pizarro.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke University)
DTSTART;VALUE=DATE-TIME:20200709T150000Z
DTEND;VALUE=DATE-TIME:20200709T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/17
DESCRIPTION:Title: On Bourgain’s counterexample for the Schrödinger maximal function
\nby Lillian Pierce (Duke University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nThere is a long and visible history of applications of
analytic methods to number theory. More recently we are starting to recog
nize applications of number-theoretic methods to analysis. In this talk we
will describe an important recent application in this direction. \n\nIn 1
980\, Carleson asked a question in PDE's: for what class of initial data f
unctions does a pointwise a.e. convergence result hold for the solution of
the linear Schrödinger equation? Over the next decades\, many people dev
eloped counterexamples to show “necessary conditions\,” and on the oth
er hand positive results to show “sufficient conditions.” In 2016 Bour
gain wrote a 3-page paper using facts from number theory to construct a fa
mily of counterexamples. A 2019 Annals paper of Du and Zhang then resolved
the question by proving positive results that push the “sufficient cond
itions” to meet Bourgain’s “necessary conditions."\n\nBourgain’s c
onstruction was regarded as somewhat mysterious. In this talk\, we give an
overview of how to rigorously derive Bourgain’s construction using idea
s from number theory. Our strategy is to start from “zero knowledge" and
gradually optimize the set-up to arrive at the final counterexample. This
talk will be broadly accessible.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bas Edixhoven (Leiden University)
DTSTART;VALUE=DATE-TIME:20200526T080000Z
DTEND;VALUE=DATE-TIME:20200526T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/18
DESCRIPTION:Title: Geometric quadratic Chabauty\nby Bas Edixhoven (Leiden Universit
y) as part of Number Theory Web Seminar\n\n\nAbstract\nJoint work with Gui
do Lido (see arxiv preprint). Determining all rational points on a curve o
f genus at least $2$ can be difficult. Chabauty's method (1941) is to inte
rsect\, for a prime number p\, in the p-adic Lie group of $p$-adic points
of the jacobian\, the closure of the Mordell-Weil group with the p-adic po
ints of the curve. If the Mordell-Weil rank is less than the genus then th
is method has never failed. Minhyong Kim's non-abelian Chabauty programme
aims to remove the condition on the rank. The simplest case\, called quadr
atic Chabauty\, was developed by Balakrishnan\, Dogra\, Mueller\, Tuitman
and Vonk\, and applied in a tour de force to the so-called cursed curve (r
ank and genus both $3$). Our work gives a version of this method that uses
only `simple algebraic geometry' (line bundles over the jacobian and mode
ls over the integers). For the talk\, no knowledge of all this algebraic g
eometry is required\, it will be accessible to all number theorists.\n\nRe
ferences: https://arxiv.org/abs/1910.10752\nArizona Winter School 2020: ht
tp://swc.math.arizona.edu/index.html\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph H. Silverman (Brown University)
DTSTART;VALUE=DATE-TIME:20200730T150000Z
DTEND;VALUE=DATE-TIME:20200730T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/19
DESCRIPTION:Title: More Tips on Keeping Secrets in a Post-Quantum World: Lattice-Based
Cryptography\nby Joseph H. Silverman (Brown University) as part of Num
ber Theory Web Seminar\n\n\nAbstract\nWhat do internet commerce\, online b
anking\, and updates to your phone apps have in common? All of them depen
d on modern public key cryptography for security. For example\, there is
the RSA cryptosystem that is used by many internet browsers\, and there i
s the elliptic curve based ECDSA digital signature scheme that is used in
many applications\, including Bitcoin. All of these cryptographic const
ruction are doomed if/when someone (NSA? Russia? China?) builds a full-
scale operational quantum computer. It hasn't happened yet\, as far as we
know\, but there are vast resources being thrown at the problem\, and sl
ow-but-steady progress is being made. So the search is on for cryptograph
ic algorithms that are secure against quantum computers. The first part
of my talk will be a mix of math and history and prognostication centere
d around the themes of quantum computers and public key cryptography. The
second part will discuss cryptographic constructions based on hard latti
ce problems\, which is one of the approaches being proposed to build a po
st-quantum cryptographic infrastructure.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20200602T080000Z
DTEND;VALUE=DATE-TIME:20200602T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/20
DESCRIPTION:Title: Multiplicative functions in short intervals revisited\nby Kaisa
Matomäki (University of Turku) as part of Number Theory Web Seminar\n\n\n
Abstract\nA few years ago Maksym Radziwill and I showed that the average o
f a multiplicative function in almost all very short intervals $[x\, x+h]$
is close to its average on a long interval $[x\, 2x]$. This result has si
nce been utilized in many applications.\n\nIn a work in progress that I wi
ll talk about\, Radziwill and I revisit the problem and generalise our res
ult to functions which vanish often as well as prove a power-saving upper
bound for the number of exceptional intervals (i.e. we show that there are
$O(X/h^\\kappa)$ exceptional $x \\in [X\, 2X]$). \n\nWe apply this result
for instance to studying gaps between norm forms of an arbitrary number f
ield.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20200806T150000Z
DTEND;VALUE=DATE-TIME:20200806T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/21
DESCRIPTION:Title: Tetrahedra with rational dihedral angles\nby Bjorn Poonen (MIT)
as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1895\, Hill discove
red a 1-parameter family of tetrahedra whose dihedral angles are all ratio
nal multiples of $\\pi$. In 1976\, Conway and Jones related the problem of
finding all such tetrahedra to solving a polynomial equation in roots of
unity. Many previous authors have solved polynomial equations in roots of
unity\, but never with more than $12$ monomials\, and the Conway-Jones pol
ynomial has $105$ monomials! I will explain the method we use to solve it
and our discovery that the full classification consists of two $1$-paramet
er families and an explicit finite list of sporadic tetrahedra.\n\nBuildin
g on this work\, we classify all configurations of vectors in $\\R^3$ such
that the angle between each pair is a rational multiple of $\\pi$. Sample
result: Ignoring trivial families and scalar multiples\, any configuratio
n with more than $9$ vectors is contained in a particular $15$-vector conf
iguration. \n\nThis is joint work with Kiran Kedlaya\, Alexander Kolpakov
\, and Michael Rubinstein.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Andrés Helfgott (Göttingen/CNRS (IMJ))
DTSTART;VALUE=DATE-TIME:20200616T080000Z
DTEND;VALUE=DATE-TIME:20200616T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/22
DESCRIPTION:Title: Optimality of the logarithmic upper-bound sieve\, with explicit esti
mates\nby Harald Andrés Helfgott (Göttingen/CNRS (IMJ)) as part of N
umber Theory Web Seminar\n\n\nAbstract\nAt the simplest level\, an upper b
ound sieve of Selberg type is a choice of $\\rho(d)$\, $d\\le D$\, with $\
\rho(1)=1$\, such that\n$$\nS = \\sum_{n\\leq N} \\left(\\sum_{d|n} \\mu(d
) \\rho(d)\\right)^2\n$$\nis as small as possible.\n\nThe optimal choice o
f $\\rho(d)$ for given $D$ was found by Selberg. However\, for several app
lications\, it is better to work with functions $\\rho(d)$ that are scalin
gs of a given continuous or monotonic function $\\eta$. The question is th
en what is the best function $\\eta$\, and how does $S$ for given $\\eta$
and $D$ compares to $S$ for Selberg's choice.\n\nThe most common choice of
eta is that of Barban-Vehov (1968)\, which gives an $S$ with the same mai
n term as Selberg's $S$. We show that Barban and Vehov's choice is optimal
among all $\\eta$\, not just (as we knew) when it comes to the main term\
, but even when it comes to the second-order term\, which is negative and
which we determine explicitly.\n\nThis is joint work with Emanuel Carneiro
\, Andrés Chirre and Julian Mejía-Cordero.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Skinner (Princeton University)
DTSTART;VALUE=DATE-TIME:20200820T150000Z
DTEND;VALUE=DATE-TIME:20200820T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/23
DESCRIPTION:Title: Solving diagonal diophantine equations over general $p$-adic fields<
/a>\nby Christopher Skinner (Princeton University) as part of Number Theor
y Web Seminar\n\n\nAbstract\nThis talk will explain a proof that a system
of $r$ diagonal equations\n$$\na_{i\,1}x_1^d + \\cdots +a_{i\,s} x_s^d = 0
\,\\quad i = 1\,...\,r\n$$\nwith coefficients in a $p$-adic field $K$ has
a non-trivial solution in $K$ if the number of variables $s$ exceeds $3r^
2d^2$ (if $p > 2$) or $8r^2d^2$ (if $p=2$). This is the first bound that
holds uniformly for all $p$-adic fields K and that is polynomial in $r$ or
$d$. The methods -- and talk -- are elementary.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hector Pasten (Pontificia Universidad Católica de Chile)
DTSTART;VALUE=DATE-TIME:20200827T150000Z
DTEND;VALUE=DATE-TIME:20200827T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/24
DESCRIPTION:Title: A Chabauty-Coleman bound for hyperbolic surfaces in abelian threefol
ds\nby Hector Pasten (Pontificia Universidad Católica de Chile) as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nA celebrated result of Cole
man gives a completely explicit version of Chabauty's finiteness theorem f
or rational points in hyperbolic curves over a number field\, by a study o
f zeros of p-adic analytic functions. After several developments around th
is result\, the problem of proving an analogous explicit bound for higher
dimensional subvarieties of abelian varieties remains elusive. In this tal
k I'll sketch the proof of such a bound for hyperbolic surfaces contained
in abelian threefolds. This is joint work with Jerson Caro.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Özlem Imamoglu (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20200917T150000Z
DTEND;VALUE=DATE-TIME:20200917T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/25
DESCRIPTION:Title: A class number formula of Hurwitz\nby Özlem Imamoglu (ETH Züri
ch) as part of Number Theory Web Seminar\n\n\nAbstract\nIn a little known
paper Hurwitz gave an infinite series representation for the class numb
er of positive definite binary quadratic forms In this talk I will report
on joint work with W. Duke and A. Toth where we show how the ideas of
Hurwitz can be applied in other settings\, in particular to give a formula
for the class number of binary cubic forms.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Breuillard (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20200924T150000Z
DTEND;VALUE=DATE-TIME:20200924T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/26
DESCRIPTION:Title: A subspace theorem for manifolds\nby Emmanuel Breuillard (Univer
sity of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstract\nIn
the late 90's Kleinbock and Margulis solved a long-standing conjecture due
to Sprindzuk regarding diophantine approximation on submanifolds of $\\R^
n$. Their method used homogeneous dynamics via the so-called non-divergenc
e estimates for unipotent flows on the space of lattices. In this talk I w
ill explain how these ideas\, combined with a certain understanding of the
geometry at the heart of Schmidt's subspace theorem\, in particular the n
otion of Harder-Narasimhan filtration\, leads to a metric version of the s
ubspace theorem\, where the linear forms are allowed to depend on a parame
ter. This subspace theorem for manifolds allows to quickly compute certain
diophantine exponents\, and it leads to several generalizations of the Kl
einbock-Margulis results in a variety of contexts. Joint work with Nicolas
de Saxcé.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART;VALUE=DATE-TIME:20200910T150000Z
DTEND;VALUE=DATE-TIME:20200910T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/27
DESCRIPTION:Title: Existence of quadratic points on intersections of quadrics\nby B
ianca Viray (University of Washington) as part of Number Theory Web Semina
r\n\n\nAbstract\nSpringer's theorem and the Amer-Brumer theorem together i
mply that intersections of two quadrics have a rational point if and only
if they have a $0$-cycle of degree $1$. In this talk\, we consider wheth
er this statement can be strengthened in the case when there is no rationa
l point\, namely whether 1) the least degree of a $0$-cycle can be bounded
\, and 2) whether there is an effective $0$-cycle of this degree. We rep
ort on results in this direction\, paying particular attention to the case
of local and global fields. This is joint work with Brendan Creutz.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART;VALUE=DATE-TIME:20200813T150000Z
DTEND;VALUE=DATE-TIME:20200813T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/28
DESCRIPTION:Title: Practical numbers\nby Carl Pomerance (Dartmouth College) as part
of Number Theory Web Seminar\n\n\nAbstract\nA practical number $n$ is one
where each number up to $n$ can be expressed as a subset sum of $n$'s pos
itive divisors. It seems that Fibonacci was interested in them since they
have the property that all fractions $m/n$ with $m < n$ can be written as
a sum of distinct unit fractions with denominators dividing $n$. With sim
ilar considerations in mind\, Srinivasan in 1948 coined the term "practica
l". There has been quite a lot of effort to study their distribution\, eff
ort which has gone hand in hand with the development of the anatomy of int
egers. After work of Tenenbaum\, Saias\, and Weingartner\, we now know th
e "Practical Number Theorem": the number of practical numbers up to $x$ is
asymptotically $cx/log x$\, where $c= 1.33607...$. In this talk I'll dis
cuss some recent developments\, including work of Thompson who considered
the allied concept of $\\phi$-practical numbers $n$ (the polynomial $t^n-1
$ has divisors over the integers of every degree up to $n$) and the proof
(joint with Weingartner) of a conjecture of Margenstern that each large od
d number can be expressed as a sum of a prime and a practical number.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:René Schoof (Università di Roma “Tor Vergata”)
DTSTART;VALUE=DATE-TIME:20200707T080000Z
DTEND;VALUE=DATE-TIME:20200707T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/29
DESCRIPTION:Title: Abelian varieties over ${\\bf Q}(\\sqrt{97})$ with good reduction ev
erywhere\nby René Schoof (Università di Roma “Tor Vergata”) as p
art of Number Theory Web Seminar\n\n\nAbstract\nUnder assumption of the Ge
neralized Riemann Hypothesis we show that every abelian variety over ${\\b
f Q}(\\sqrt{97})$ with good reduction everywhere is isogenous to a power o
f a certain $3$-dimensional modular abelian variety.\n\n(joint with Lassin
a Dembele)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kannan Soundararajan (Stanford University)
DTSTART;VALUE=DATE-TIME:20200630T000000Z
DTEND;VALUE=DATE-TIME:20200630T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/30
DESCRIPTION:Title: Equidistribution from the Chinese Remainder Theorem\nby Kannan S
oundararajan (Stanford University) as part of Number Theory Web Seminar\n\
n\nAbstract\nSuppose for each prime $p$ we are given a set $A_p$ (possibly
empty) of residue classes mod $p$. Use these and the Chinese Remainder T
heorem to form a set $A_q$ of residue classes mod $q$\, for any integer $q
$. Under very mild hypotheses\, we show that for a typical integer $q$\,
the residue classes in $A_q$ will become equidistributed. The prototypica
l example (which this generalises) is Hooley's theorem that the roots of a
polynomial congruence mod $n$ are equidistributed on average over $n$. I
will also discuss generalisations of such results to higher dimensions\,
and when restricted to integers with a given number of prime factors. (Jo
int work with Emmanuel Kowalski.)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART;VALUE=DATE-TIME:20200723T150000Z
DTEND;VALUE=DATE-TIME:20200723T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/31
DESCRIPTION:Title: What’s up in arithmetic statistics\nby Jordan Ellenberg (Unive
rsity of Wisconsin–Madison) as part of Number Theory Web Seminar\n\n\nAb
stract\nIf not for a global pandemic\, a bunch of mathematicians would hav
e gathered in Germany to talk about what’s going on in the geometry of a
rithmetic statistics\, which I would roughly describe as “methods from a
rithmetic geometry brought to bear on probabilistic questions about arithm
etic objects". What does the maximal unramified extension of a random numb
er field look like? What is the probability that a random elliptic curve h
as a $2$-Selmer group of rank 100? How do you count points on a stack? I
’ll give a survey of what’s happening in questions in this area\, tryi
ng to emphasize open questions.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ken Ono (University of Virginia)
DTSTART;VALUE=DATE-TIME:20200714T000000Z
DTEND;VALUE=DATE-TIME:20200714T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/32
DESCRIPTION:Title: Variants of Lehmer's speculation for newforms\nby Ken Ono (Unive
rsity of Virginia) as part of Number Theory Web Seminar\n\n\nAbstract\nIn
the spirit of Lehmer's unresolved speculation on the nonvanishing of Raman
ujan's tau-function\, it is natural to ask whether a fixed integer is a va
lue of τ(n)\, or is a Fourier coefficient of any given newform. In joint
work with J. Balakrishnan\, W. Craig\, and W.-L. Tsai\, the speaker has o
btained some results that will be described here. For example\, infinitely
many spaces are presented for which the primes ℓ≤37 are not absolute
values of coefficients of any newforms with integer coefficients. For Rama
nujan’s tau-function\, such results imply\, for n>1\, that\n\nτ(n)∉{
±1\,±3\,±5\,±7\,±13\,±17\,−19\,±23\,±37\,±691}.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wadim Zudilin (Radboud University Nijmegen)
DTSTART;VALUE=DATE-TIME:20200721T080000Z
DTEND;VALUE=DATE-TIME:20200721T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/33
DESCRIPTION:Title: Irrationality through an irrational time\nby Wadim Zudilin (Radb
oud University Nijmegen) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nAfter reviewing some recent development and achievements related to dio
phantine problems of the values of Riemann's zeta function and generalized
polylogarithms (not all coming from myself!)\, I will move the focus to $
\\pi=3.1415926\\dots$ and its rational approximations. Specifically\, I wi
ll discuss a construction of rational approximations to the number that le
ads to the record irrationality measure of $\\pi$. The talk is based on jo
int work with Doron Zeilberger.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Ford (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20200903T150000Z
DTEND;VALUE=DATE-TIME:20200903T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/34
DESCRIPTION:Title: Prime gaps\, probabilistic models\, the interval sieve\, Hardy-Littl
ewood conjectures and Siegel zeros\nby Kevin Ford (University of Illin
ois at Urbana-Champaign) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nMotivated by a new probabilistic interpretation of the Hardy-Littlewood
$k$-tuples conjectures\, we introduce a new probabilistic model of the pr
imes and make a new conjecture about the largest gaps between the primes b
elow $x$. Our bound depends on a property of the interval sieve which is n
ot well understood. We also show that any sequence of integers which satis
fies a sufficiently uniform version of the Hardy-Littlewood conjectures mu
st have large gaps of a specific size. Finally\, assuming that Siegel zero
s exist we show the existence of gaps between primes which are substantial
ly larger than the gaps which are known unconditionally. Much of this work
is joint with Bill Banks and Terry Tao.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Ho (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201001T150000Z
DTEND;VALUE=DATE-TIME:20201001T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/35
DESCRIPTION:Title: The Hasse local-to-global principle for some genus one curves\nb
y Wei Ho (University of Michigan) as part of Number Theory Web Seminar\n\n
\nAbstract\nThe Hasse principle is a useful guiding philosophy in arithmet
ic geometry that relates "global" questions to analogous "local" questions
\, which are often easier to understand. A simple incarnation of the Hasse
principle says that a given polynomial equation has a solution in the rat
ional numbers (i.e.\, is "globally soluble") if and only if it has a solut
ion in the real numbers and in the p-adic numbers for all primes p (i.e.\,
is "everywhere locally soluble"). While this principle holds for many "si
mple" such polynomials\, it is a very difficult question to classify the p
olynomials (or more generally\, algebraic varieties) for which the princip
le holds or fails.\n\nIn this talk\, we will discuss problems related to t
he Hasse principle for some classes of varieties\, with a special focus on
genus one curves given by bihomogeneous polynomials of bidegree $(2\,2)$
in $\\mathbb{P}^1 \\times \\mathbb{P}^1$. For example\, we will describe h
ow to compute the proportion of these curves that are everywhere locally s
oluble (joint work with Tom Fisher and Jennifer Park)\, and we will explai
n why the Hasse principle fails for a positive proportion of these curves\
, by comparing the average sizes of $2$- and $3$-Selmer groups for a famil
y of elliptic curves with a marked point (joint work with Manjul Bhargava)
.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (EPFL)
DTSTART;VALUE=DATE-TIME:20201008T150000Z
DTEND;VALUE=DATE-TIME:20201008T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/36
DESCRIPTION:Title: Simultaneous reductions of CM elliptic curves\nby Philippe Miche
l (EPFL) as part of Number Theory Web Seminar\n\n\nAbstract\nLet $E$ be an
elliptic curve with CM by the imaginary quadratic order $O_D$ of discrimi
nant $D<0$. Given $p$ a prime \; if $p$ is inert or ramified in the quadra
tic field generated by $\\sqrt D$ then $E$ has supersingular reduction at
a(ny) fixed place above $p$. By a variant of Duke’s equidistribution the
orem\, as $D$ grows along such discriminants\, the proportion of CM ellipt
ic curves with CM by $O_D$ whose reduction at such place is a given supers
ingular curve converge to a natural (non-zero) limit. A further step is to
fix several (distinct) primes $p_1\,\\cdots\,p_s$ and to look for the pro
portion of CM curves whose reduction above each of these primes is prescri
bed. In this talk\, we will explain how a powerful result of Einsiedler an
d Lindenstrauss classifying joinings of rank $2$ actions on products of lo
cally homogeneous spaces implies that as $D$ grows along adequate subseque
nces of negative discriminants\, this proportion converge to the product o
f the limits for each individual $p_i$ (a sort of asymptotic Chinese Remin
der Theorem for reductions of CM elliptic curves if you wish). This is joi
nt work with M. Aka\, M. Luethi and A.Wieser. If time permits\, we will al
so describe a further refinement -- obtained with the additional collabora
tion of R. Menares — of these equidistribution results for the formal gr
oups attached to these curves.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore Pisa)
DTSTART;VALUE=DATE-TIME:20200901T090000Z
DTEND;VALUE=DATE-TIME:20200901T100000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/37
DESCRIPTION:Title: Torsion in elliptic familes and applications to billiards\nby Um
berto Zannier (Scuola Normale Superiore Pisa) as part of Number Theory Web
Seminar\n\n\nAbstract\nWe shall consider elliptic pencils\, of which the
best-known example is probably the Legendre family $L_t$: $y^2=x(x-1)(x-t)
$ where $t$ is a parameter. Given a section $P(t)$ (i.e. a family of poin
ts on $L_t$ depending on $t$) it is an issue to study the set of complex
$b$ such that $P(b)$ is torsion on $L_b$. We shall recall a number of resu
lts on the nature of this set. Then we shall present some applications (ob
tained jointly with P. Corvaja) to elliptical billiards. For instance\, if
two players hit the same ball with directions forming a given angle in $(
0\,\\pi)$\, there are only finitely many cases for which both billiard tra
jectories are periodic.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron L. Stewart (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20201015T150000Z
DTEND;VALUE=DATE-TIME:20201015T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/38
DESCRIPTION:Title: On integers represented by binary forms\nby Cameron L. Stewart (
University of Waterloo) as part of Number Theory Web Seminar\n\n\nAbstract
\nWe shall discuss the following results which are joint work with Stanley
Xiao.\n\nLet $F(x\,y)$ be a binary form with integer coefficients\, degre
e $d(>2)$ and non-zero discriminant. There is a positive number $C(F)$ suc
h that the number of integers of absolute value at most $Z$ which are repr
esented by $F$ is asymptotic to $C(F)Z^{2/d}$.\n\nLet $k$ be an integer wi
th $k>1$ and suppose that there is no prime $p$ such that $p^k$ divides $F
(a\,b)$ for all pairs of integers $(a\,b)$. Then\, provided that $k$ excee
ds $7d/18$ or $(k\,d)$ is $(2\,6)$ or $(3\,8)$\, there is a positive numbe
r $C(F\,k)$ such that the number of $k$-free integers of absolute value at
most $Z$ which are represented by $F$ is asymptotic to $C(F\,k)Z^{2/d}$.\
n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Konyagin (Steklov Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20201022T150000Z
DTEND;VALUE=DATE-TIME:20201022T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/39
DESCRIPTION:Title: A construction of A. Schinzel - many numbers in a short interval wit
hout small prime factors\nby Sergei Konyagin (Steklov Institute of Mat
hematics) as part of Number Theory Web Seminar\n\n\nAbstract\nHardy and Li
ttlewood (1923) conjectured that for any integers $x\,y\\ge2$\n$$\n\\pi(x+
y) \\le \\pi(x) + \\pi(y). \\qquad\\qquad\\qquad (1)\n$$\n\nLet us call a
set $\\{b_1\,\\dots\,b_k\\}$ of integers admissible if for each\nprime $p$
there is some congruence class $\\bmod p$ which contains none\nof the int
egers $b_i$. The prime $k$-tuple conjecture states that if a set \n$\\{b_1
\,\\dots\,b_k\\}$ is admissible\, then there exist infinitely many \ninteg
ers $n$ for which all the numbers $n+b_1\,\\dots\,n+b_k$ are primes.\n\nLe
t $x$ be a positive integer and $\\rho^*(x)$ be the maximum number\nof int
egers in any interval $(y\,y+x]$ (with no restriction on $y$)\nwhich are r
elatively prime to all positive integers $\\le x$.\nThe prime $k$-tuple co
njecture implies that\n$$\\max_{y\\ge x}(\\pi(x+y)-\\pi(y))=\\limsup_{y\\g
e x} (\\pi(x+y)-\\pi(y))=\\rho^*(x).$$\n\nHensley and Richards (1974) prov
ed that\n$$\\rho^*(x) - \\pi(x) \\ge(\\log 2- o(1)) x(\\log x)^{-2}\\quad(
x\\to\\infty).$$\nTherefore\, (1) is not compatible with the prime $k$-tup
le\nconjecture. Using a construction of Schinzel we show that\n$$\\rho^*(x
) - \\pi(x) \\ge((1/2)- o(1)) x(\\log x)^{-2}\\log\\log\\log x\\quad(x\\to
\\infty).$$\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dzmitry Badziahin (University of Sydney)
DTSTART;VALUE=DATE-TIME:20200915T090000Z
DTEND;VALUE=DATE-TIME:20200915T100000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/40
DESCRIPTION:Title: Approximation by algebraic numbers\nby Dzmitry Badziahin (Univer
sity of Sydney) as part of Number Theory Web Seminar\n\n\nAbstract\nIn thi
s talk we discuss the approximation of transcendental numbers by algebraic
numbers of given degree and bounded height. More precisely\, for any real
number $x$\, by $w_n^*(x)$ we define the supremum of all positive real va
lues $w$ such that the inequality\n $$ |x - a| < H(a)^{-w-1}$$\nhas
infinitely many solutions in algebraic real numbers $a$ of degree at most
$n$. Here $H(a)$ means the naive height of the minimal polynomial in $\\Z[
x]$ with coprime coefficients. In 1961\, Wirsing asked: is it true that th
e quantity $w_n^*(x)$ is at least n for all transcendental $x$? Apart from
partial results for small values of $n$\, this problem still remains open
. Wirsing himself managed to establish the lower bound of the form $w_n^*(
x) \\ge n/2+1 - o(1)$. Until recently\, the only improvements to this boun
d were in terms of $O(1)$. I will talk about our resent work with Schleisc
hitz where we managed to improve the bound by a quantity of the size $O(n)
$. More precisely\, we show that $w_n^*(x) > n/\\sqrt{3}$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART;VALUE=DATE-TIME:20201029T160000Z
DTEND;VALUE=DATE-TIME:20201029T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/41
DESCRIPTION:Title: The distribution of prime polynomials over finite fields\nby Wil
l Sawin (Columbia University) as part of Number Theory Web Seminar\n\n\nAb
stract\nMany conjectures in number theory have analogues for polynomials i
n one variable over a finite field. In recent works with Mark Shusterman\,
we proved analogues of two conjectures about prime numbers - the twin pri
mes conjecture and the conjecture that there are infinitely many primes of
the form $n^2+1$. I will describe these results and explain some of the k
ey ideas in the proofs\, which combine classical analytic methods\, elemen
tary algebraic manipulations\, and geometric methods.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dragos Ghioca (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20201201T010000Z
DTEND;VALUE=DATE-TIME:20201201T020000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/42
DESCRIPTION:Title: A couple of conjectures in arithmetic dynamics over fields of positi
ve characteristic\nby Dragos Ghioca (University of British Columbia) a
s part of Number Theory Web Seminar\n\n\nAbstract\nThe Dynamical Mordell-L
ang Conjecture predicts the structure of the intersection between a subvar
iety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ w
ith the orbit of a point in $X(K)$ under an endomorphism $\\Phi$ of $X$. T
he Zariski dense conjecture provides a dichotomy for any rational self-map
$\\Phi$ of a variety $X$ defined over an algebraically closed field $K$ o
f characteristic $0$: either there exists a point in $X(K)$ with a well-de
fined Zariski dense orbit\, or $\\Phi$ leaves invariant some non-constant
rational function $f$. For each one of these two conjectures we formulate
an analogue in characteristic $p$\; in both cases\, the presence of the Fr
obenius endomorphism in the case $X$ is isotrivial creates significant com
plications which we will explain in the case of algebraic tori.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya D. Shkredov (Steklov Mathematical Institute\, Moscow)
DTSTART;VALUE=DATE-TIME:20200922T090000Z
DTEND;VALUE=DATE-TIME:20200922T100000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/43
DESCRIPTION:Title: Zaremba's conjecture and growth in groups\nby Ilya D. Shkredov (
Steklov Mathematical Institute\, Moscow) as part of Number Theory Web Semi
nar\n\n\nAbstract\nZaremba's conjecture belongs to the area of continued f
ractions. It predicts that for any given positive integer $q$ there is a p
ositive $a$\, $a < q$\, $(a\,q)=1$ such that all partial quotients $b_j$
in its continued fractions expansion $a/q = 1/b_1+1/b_2 +...+ 1/b_s$ are b
ounded by five. At the moment the question is widely open although the are
a has a rich history of works by Korobov\, Hensley\, Niederreiter\, Bourga
in and many others. We survey certain results concerning this hypothesis a
nd show how growth in groups helps to solve different relaxations of Zarem
ba's conjecture. In particular\, we show that a deeper hypothesis of Hensl
ey concerning some Cantor-type set with the Hausdorff dimension $>1/2$ tak
es place for the so-called modular form of Zaremba's conjecture.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Harvard University)
DTSTART;VALUE=DATE-TIME:20201006T000000Z
DTEND;VALUE=DATE-TIME:20201006T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/44
DESCRIPTION:Title: Selmer groups and a Cassels-Tate pairing for finite Galois modules\nby Alexander Smith (Harvard University) as part of Number Theory Web S
eminar\n\n\nAbstract\nI will discuss some new results on the structure of
Selmer groups of finite Galois modules over global fields. Tate's definiti
on of the Cassels-Tate pairing can be extended to a pairing on such Selmer
groups with little adjustment\, and many of the fundamental properties of
the Cassels-Tate pairing can be reproved with new methods in this setting
. I will also give a general definition of the theta/Mumford group and rel
ate it to the structure of the Cassels-Tate pairing\, generalizing work of
Poonen and Stoll.\n\nAs one application of this theory\, I will prove an
elementary result on the symmetry of the class group pairing for number fi
elds with many roots of unity and connect this to the work of mine and oth
ers on class group statistics.\n\nThis work is joint with Adam Morgan.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Tzu-Yueh Wang (Academia Sinica)
DTSTART;VALUE=DATE-TIME:20200929T000000Z
DTEND;VALUE=DATE-TIME:20200929T010000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/45
DESCRIPTION:Title: Pisot's $d$-th root's conjecture for function fields and its complex
analog\nby Julie Tzu-Yueh Wang (Academia Sinica) as part of Number Th
eory Web Seminar\n\n\nAbstract\nPisot's $d$-th root's conjecture\, proved
by Zannier in 2000\, can be stated as follows.\nLet $b$ be a linear rec
urrence \nover a number field $k$\, and $d\\ge2$ be an integer. Suppose t
hat\n$b(n)$ is the $d$-th power of some element in $k$ for all but finitel
y\nmany $n$. Then there exists a linear recurrence $a$\nover $\\overline{k
}$ such that $a(n)^{d}=b(n)$ for all $n$.\n\n\nIn this talk\, we propose
a function-field analog of this result and prove it under some ``non-triv
iality''\nassumption. We relate the problem to a result of Pasten-Wang
on B\\"uchi's $d$-th power problem and develop a function-field GCD est
imate for multivariable polynomials with ``small coefficients" evaluating
at $S$-units arguments. We will also discuss its complex analog in the no
tion of (generalized Ritt's) exponential polynomials. \n\nThis is a jo
int work with Ji Guo and Chia-Liang Sun.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maryna Viazovska (EPFL)
DTSTART;VALUE=DATE-TIME:20200908T080000Z
DTEND;VALUE=DATE-TIME:20200908T090000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/46
DESCRIPTION:Title: Universal optimality\, Fourier interpolation\, and modular integrals
\nby Maryna Viazovska (EPFL) as part of Number Theory Web Seminar\n\n\
nAbstract\nIn this lecture we will show that the E8 and Leech lattices mi
nimize energy for a wide class of potential functions. This theorem implie
s recently proven optimality of E8 and Leech lattices as sphere packings a
nd broadly generalizes it to long-range interactions. The key ingredient o
f the proof is sharp linear programming bounds. Construction of the optima
l auxiliary functions attaining these bounds is based on a new interpolati
on theorem. This is joint work with Henry Cohn\, Abhinav Kumar\, Stephen D
. Miller\, and Danylo Radchenko.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART;VALUE=DATE-TIME:20201105T160000Z
DTEND;VALUE=DATE-TIME:20201105T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/47
DESCRIPTION:Title: Distribution of lattice points on hyperbolic circles\nby Par Kur
lberg (KTH) as part of Number Theory Web Seminar\n\n\nAbstract\nWe study t
he distribution of lattice points lying on expanding circles in the hyperb
olic plane. The angles of lattice points arising from the orbit of the mod
ular group $\\mathrm{PSL}(2\,\\Z)$\, and lying on hyperbolic circles cente
red at i\, are shown to be equidistributed for generic radii (among the on
es that contain points). We also show that angles fail to equidistribute o
n a thin set of exceptional radii\, even in the presence of growing multip
licity. Surprisingly\, the distribution of angles on hyperbolic circles tu
rns out to be related to the angular distribution of euclidean lattice poi
nts lying on circles in the plane\, along a thin subsequence of radii. Thi
s is joint work with D. Chatzakos\, S. Lester and I. Wigman.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Gorodnik (University of Zurich)
DTSTART;VALUE=DATE-TIME:20201013T090000Z
DTEND;VALUE=DATE-TIME:20201013T100000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/49
DESCRIPTION:Title: Arithmetic approach to the spectral gap problem\nby Alexander Go
rodnik (University of Zurich) as part of Number Theory Web Seminar\n\n\nAb
stract\nThe spectral gap is an analytic property of group actions which ca
n be described as absence of "almost invariant vectors" or more quantitati
vely in terms of norm bounds for suitable averaging operators. In the sett
ing of homogeneous spaces this property also has a profound number-theoret
ic meaning since it is closely related to understanding the automorphic re
presentations. In this talk we survey some previous results about the spec
tral gap property and describe new approaches to deriving upper and lower
bounds for the spectral gap.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia University)
DTSTART;VALUE=DATE-TIME:20201119T160000Z
DTEND;VALUE=DATE-TIME:20201119T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/50
DESCRIPTION:Title: CANCELLED--Moments and non-vanishing of cubic Dirichlet $L$-function
s at $s=\\frac{1}{2}$\nby Chantal David (Concordia University) as part
of Number Theory Web Seminar\n\n\nAbstract\nA famous conjecture of Chowla
predicts that $L(\\frac{1}{2}\,\\chi)\\ne 0$ for all Dirichlet $L$-functi
ons\nattached to primitive characters $\\chi$. It was conjectured first in
the case where $\\chi$ is a quadratic\ncharacter\, which is the most stud
ied case. For quadratic Dirichlet $L$-functions\, Soundararajan\nproved th
at at least 87.5% of the quadratic Dirichlet $L$-functions do not vanish a
t $s=\\frac{1}{2}$.\nUnder GRH\, there are slightly stronger results by Oz
lek and Snyder.\n\nWe present in this talk the first result showing a posi
tive proportion of cubic Dirichlet\n$L$-functions non-vanishing at $s=\\fr
ac{1}{2}$ for the non-Kummer case over function fields. This can\nbe achie
ved by using the recent breakthrough work on sharp upper bounds for moment
s of\nSoundararajan\, Harper and Lester-Radziwill. Our results would trans
fer over number fields\,\nbut we would need to assume GRH in this case.\n\
nCANCELLED! There will be no talk this Thursday.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksym Radziwill (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20201210T160000Z
DTEND;VALUE=DATE-TIME:20201210T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/51
DESCRIPTION:Title: The Fyodorov-Hiary-Keating conjecture\nby Maksym Radziwill (Cali
fornia Institute of Technology) as part of Number Theory Web Seminar\n\n\n
Abstract\nI will discuss recent progress on the Fyodorov-Hiary-Keating con
jecture on the distribution of the local maximum of the Riemann zeta-funct
ion. This is joint work with Louis-Pierre Arguin and Paul Bourgade.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Masser (University of Basel)
DTSTART;VALUE=DATE-TIME:20201112T160000Z
DTEND;VALUE=DATE-TIME:20201112T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/52
DESCRIPTION:Title: Pencils of norm form equations and a conjecture of Thomas\nby Da
vid Masser (University of Basel) as part of Number Theory Web Seminar\n\n\
nAbstract\nWe consider certain one-parameter families of norm form (and ot
her) diophantine equations\, and we solve them completely and uniformly fo
r all sufficiently large positive integer values of the parameter (everyth
ing effective)\, following a line started by Emery Thomas in 1990. The new
tool is a bounded height result from 2017 by Francesco Amoroso\, Umberto
Zannier and the speaker.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jörg Brüdern (University of Göttingen)
DTSTART;VALUE=DATE-TIME:20201020T090000Z
DTEND;VALUE=DATE-TIME:20201020T100000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/53
DESCRIPTION:Title: Harmonic analysis of arithmetic functions\nby Jörg Brüdern (Un
iversity of Göttingen) as part of Number Theory Web Seminar\n\n\nAbstract
\nWe study arithmetic functions that are bounded in mean square\, and simu
ltaneously have a mean value over any arithmetic progression. A Besicovitc
h type norm makes the set of these functions a Banach space. We apply the
Hardy-Littlewood (circle) method to analyse this space. This method turns
out to be a surprisingly flexible tool for this purpose. We obtain several
characterisations of limit periodic functions\, correlation formulae\, an
d we give some applications to Waring's problem and related topics. Finall
y\, we direct the theory to the distribution of the arithmetic functions u
nder review in arithmetic progressions\, with mean square results of Barba
n-Davenport-Halberstam type and related asymptotic formulae at the focus o
f our attention. There is a rich literature on this last theme. Our approa
ch supersedes previous work in various ways\, and ultimately provides anot
her characterisation of limit periodic functions: the variance over arithm
etic progression is atypically small if and only if the input function is
limit periodic.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gal Binyamini (Weizmann Institute of Science)
DTSTART;VALUE=DATE-TIME:20201027T100000Z
DTEND;VALUE=DATE-TIME:20201027T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/54
DESCRIPTION:Title: Point counting for foliations in Diophantine geometry\nby Gal Bi
nyamini (Weizmann Institute of Science) as part of Number Theory Web Semin
ar\n\n\nAbstract\nI will discuss "point counting" in two broad senses: cou
nting the intersections between a trascendental variety and an algebraic o
ne\; and counting the number of algebraic points\, as a function of degree
and height\, on a transcendental variety. After reviewing the fundamental
results in this area - from the theory of o-minimal structures and the Pi
la-Wilkie theorem\, I will restrict attention to the case that the transce
ndental variety is given in terms of a leaf of an algebraic foliation\, an
d everything is defined over a number field. It turns out that in this cas
e far stronger estimates can be obtained.\n\nApplying the above to foliati
ons associated to principal G-bundles on various moduli spaces\, many clas
sical application of the Pila-Wilkie theorem can be sharpened and effectiv
ized. In particular I will discuss issues around effectivity and polynomia
l-time solvability for the Andre-Oort conjecture\, unlikely intersections
in abelian schemes\, and some related directions.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201207T220000Z
DTEND;VALUE=DATE-TIME:20201207T230000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/55
DESCRIPTION:Title: Bounding torsion in class group and families of local systems\nb
y Jacob Tsimerman (University of Toronto) as part of Number Theory Web Sem
inar\n\n\nAbstract\n(joint w/ Arul Shankar) We discuss a new method to bou
nd 5-torsion in class groups of quadratic fields using the refined BSD con
jecture for elliptic curves. The most natural “trivial” bound on the n
-torsion is to bound it by the size of the entire class group\, for which
one has a global class number formula. We explain how to make sense of the
n-torsion of a class group intrinsically as a selmer group of a Galois mo
dule. We may then similarly bound its size by the Tate-Shafarevich group o
f an appropriate elliptic curve\, which we can bound using the BSD conject
ure. This fits into a general paradigm where one bounds selmer groups of f
inite Galois modules by embedding into global objects\, and using class nu
mber formulas. If time permits\, we explain how the function field picture
yields unconditional results and suggests further generalizations.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gisbert Wüstholz (ETH / University Zurich)
DTSTART;VALUE=DATE-TIME:20201217T160000Z
DTEND;VALUE=DATE-TIME:20201217T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/56
DESCRIPTION:Title: Baker's theory for $1$-motives\nby Gisbert Wüstholz (ETH / Univ
ersity Zurich) as part of Number Theory Web Seminar\n\n\nAbstract\nFrom a
historical point of view transcendence theory used to be a nice\ncollecti
on of mostly particular results\, very difficult to find and to prove. To
find\nnumbers for which one has a chance to prove transcendence is very di
fficult.\nTo state conjecture is not so difficult but in most cases hopele
ss to prove.\nIn our lecture we try to draw a picture of quite far reachin
g frames in the theory\nof motives which can put transcendence theory into
a more conceptual setting.\n\nLooking at periods of rational $1$-forms on
varieties we realized that there is a\nmore conceptual background behind
the properties of these complex numbers \nthan had been thought so far. Th
e central question which I was trying for more than\nthree decades to answ
er was to determine when a period is algebraic. A priori a period is zero
\, algebraic\nor transcendental\, no surprise! It is also not difficult to
give examples for cases when periods are algebraic.\nHowever the big ques
tion was whether the examples are all examples. Quite recently\, partly jo
intly\nwith Annette Huber we developed a new transcendence theory within $
1$-motives which extend commutative algebraic groups. One outcome was that
algebraicity of periods has a very conceptual description\nand we shall
give a precise and surprisingly simple answer. \n\n Many questions which w
ere central in transcendence theory and with a long \nand famous history
turn out to get a general answer within the new theory. The classical wo
rk of Baker \nturns out to be a very special but seminal case.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (University of Bristol)
DTSTART;VALUE=DATE-TIME:20201103T100000Z
DTEND;VALUE=DATE-TIME:20201103T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/57
DESCRIPTION:Title: The three gap theorem in higher dimensions\nby Jens Marklof (Uni
versity of Bristol) as part of Number Theory Web Seminar\n\n\nAbstract\nTa
ke a point on the unit circle and rotate it N times by a fixed angle. The
N points thus generated partition the circle into N intervals. A beautiful
fact\, first conjectured by Hugo Steinhaus in the 1950s and proved indepe
ndently by Vera Sós\, János Surányi and Stanisław Świerczkowski\, is
that for any choice of N\, no matter how large\, these intervals can have
at most three distinct lengths. In this lecture I will explore an interpre
tation of the three gap theorem in terms of the space of Euclidean lattice
s\, which will produce various new results in higher dimensions\, includin
g gaps in the fractional parts of linear forms and nearest neighbour dista
nces in multi-dimensional Kronecker sequences. The lecture is based on joi
nt work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).\n
\n1. Wikipedia\, https://en.wikipedia.org/wiki/Three-gap_theorem \n\n2. J.
Marklof and A. Strömbergsson\, The three gap theorem and the space of la
ttices\, American Mathematical Monthly 124 (2017) 741-745 https://people.m
aths.bris.ac.uk/~majm/bib/threegap.pdf\n\n3. A. Haynes and J. Marklof\, Hi
gher dimensional Steinhaus and Slater problems via homogeneous dynamics\,
Annales scientifiques de l'Ecole normale superieure 53 (2020) 537-557 http
s://people.maths.bris.ac.uk/~majm/bib/steinhaus.pdf\n\n4. A. Haynes and J.
Marklof\, A five distance theorem for Kronecker sequences\, preprint arXi
v:2009.08444 https://people.maths.bris.ac.uk/~majm/bib/steinhaus2.pdf\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Stoll (University of Bayreuth)
DTSTART;VALUE=DATE-TIME:20201126T160000Z
DTEND;VALUE=DATE-TIME:20201126T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/58
DESCRIPTION:Title: An application of "Selmer group Chabauty" to arithmetic dynamics
\nby Michael Stoll (University of Bayreuth) as part of Number Theory Web S
eminar\n\n\nAbstract\nThe irreducibility or otherwise of iterates of polyn
omials is an\nimportant question in arithmetic dynamics. For example\, it
is\nconjectured that whenever the second iterate of $x^2 + c$ (with $c$ a\
nrational number) is irreducible over $\\Q$\, then so are all iterates.\n\
nA sufficient criterion for the iterates to be irreducible can be\nexpress
ed in terms of rational points on certain hyperelliptic curves.\nWe will s
how how to use the "Selmer group Chabauty" method developed by\nthe speake
r to determine the set of rational points on a hyperelliptic\ncurve of gen
us $7$. This leads to a proof that the seventh iterate of\n$x^2 + c$ must
be irreducible if the second iterate is. Assuming GRH\, we\ncan extend thi
s to the tenth iterate.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Harper (University of Warwick)
DTSTART;VALUE=DATE-TIME:20201215T100000Z
DTEND;VALUE=DATE-TIME:20201215T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/59
DESCRIPTION:Title: Large fluctuations of random multiplicative functions\nby Adam H
arper (University of Warwick) as part of Number Theory Web Seminar\n\n\nAb
stract\nRandom multiplicative functions $f(n)$ are a well studied random m
odel for deterministic multiplicative functions like Dirichlet characters
or the Mobius function. Arguably the first question ever studied about the
m\, by Wintner in 1944\, was to obtain almost sure bounds for the largest
fluctuations of their partial $\\sum_{n \\leq x} f(n)$\, seeking to emulat
e the classical Law of the Iterated Logarithm for independent random varia
bles. It remains an open question to sharply determine the size of these f
luctuations\, and in this talk I will describe a new result in that direct
ion. I hope to get to some interesting details of the new proof in the lat
ter part of the talk\, but most of the discussion should be widely accessi
ble.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Lubotzky (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20201203T160000Z
DTEND;VALUE=DATE-TIME:20201203T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/61
DESCRIPTION:Title: From Ramanujan graphs to Ramanujan complexes\nby Alexander Lubot
zky (Hebrew University of Jerusalem) as part of Number Theory Web Seminar\
n\n\nAbstract\nRamanujan graphs are $k$-regular graphs with all non trivi
al eigenvalues bounded (in absolute value) by $2\\sqrt{k-1}$. They are op
timal expanders (from spectral point of view). Explicit constructions of s
uch graphs were given in the 80's as quotients of the Bruhat-Tits tree ass
ociated with $\\GL(2)$ over a local field $F$\, by the action of suitable
congruence subgroups of arithmetic groups. The spectral bound was proved u
sing works of Hecke\, Deligne and Drinfeld on the "Ramanujan conjecture" i
n the theory of automorphic forms.\n\nThe work of Lafforgue\, extending D
rinfeld from $\\GL(2)$ to $\\GL(n)$\, opened the door for the constructio
n of Ramanujan complexes as quotients of the Bruhat-Tits buildings associa
ted with $\\GL(n)$ over $F$. This way one gets finite simplicial complex
es which on one hand are "random like" and at the same time have strong sy
mmetries. These seemingly contradicting properties make them very useful f
or constructions of various external objects. \n\nRecently various appli
cations have been found in combinatorics\, coding theory and in relation t
o Gromov's overlapping properties. We will survey some of these applicati
ons.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianya Liu (Shandong University)
DTSTART;VALUE=DATE-TIME:20201222T100000Z
DTEND;VALUE=DATE-TIME:20201222T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/62
DESCRIPTION:Title: Mobius disjointness for irregular flows\nby Jianya Liu (Shandong
University) as part of Number Theory Web Seminar\n\n\nAbstract\nThe behav
ior of the Mobius function is central in the theory of prime numbers. A su
rprising connection with the theory of dynamical systems was discovered in
2010 by P. Sarnak\, who formulated the Mobius Disjointness Conjecture (MD
C)\, which asserts that the Mobius function is linearly disjoint from any
zero-entropy flows. This conjecture opened the way into a large body of re
search on the interface of analytic number theory and ergodic theory. In t
his talk I will report how to establish MDC for a class of irregular flows
\, which are in general mysterious and ill understood. This is based on jo
int works with P. Sarnak\, and with W. Huang and K. Wang.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jasmin Matz (University of Copenhagen)
DTSTART;VALUE=DATE-TIME:20201124T100000Z
DTEND;VALUE=DATE-TIME:20201124T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/63
DESCRIPTION:Title: Quantum ergodicity of compact quotients of $SL(n\,R)/SO(n)$ in the l
evel aspect\nby Jasmin Matz (University of Copenhagen) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nSuppose $M$ is a closed Riemannian ma
nifold with an orthonormal basis $B$\nof $L^2(M)$ consisting of Laplace ei
genfunctions. A classical result of\nShnirelman and others proves that if
the geodesic flow on the cotangent\nbundle of $M$ is ergodic\, then $M$ is
quantum ergodic\, in particular\, on\naverage\, the probability measures
defined by the functions $f$ in $B$ on $M$\ntends on average towards the R
iemannian measure on $M$ in the high\nenergy limit (i.e\, as the Laplace e
igenvalues of $f \\to \\infty$). \nWe now want to look at a level aspect o
f this property\, namely\, instead\nof taking a fixed manifold and high en
ergy eigenfunctions\, we take a\nsequence of Benjamini-Schramm convergent
compact Riemannian manifolds\n$M_j$ together with Laplace eigenfunctions $
f$ whose eigenvalue varies in\nshort intervals. This perspective has been
recently studied in the\ncontext of graphs by Anantharaman and Le Masson\,
and for hyperbolic\nsurfaces and manifolds by Abert\, Bergeron\, Le Masso
n\, and Sahlsten. In\nmy talk I want to discuss joint work with F. Brumley
in which we study\nthis question in higher rank\, namely sequences of com
pact quotients of\n$SL(n\,R)/SO(n)$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gérald Tenenbaum (Université de Lorraine)
DTSTART;VALUE=DATE-TIME:20201110T100000Z
DTEND;VALUE=DATE-TIME:20201110T110000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/65
DESCRIPTION:Title: Recent progress on the Selberg-Delange method in analytic number the
ory\nby Gérald Tenenbaum (Université de Lorraine) as part of Number
Theory Web Seminar\n\n\nAbstract\nLet $\\varrho$ be a complex number and l
et $f$ be a multiplicative arithmetic function whose Dirichlet series take
s the form $\\zeta(s)^\\varrho G(s)$\, where $\\zeta(s)$ is the Riemann ze
ta function and $G$ is associated to a multiplicative function $g$. The cl
assical Selberg-Delange method furnishes asymptotic estimates for the aver
ages of $f$ under assumptions of either analytic continuation for $G$\, or
absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$
. We shall recall these statements and briefly describe the proofs. The ma
in part of of the lecture will be devoted to give an account on recent wor
ks (in particular a joint paper with Régis de la Bretèche) considering d
ifferent set of hypotheses\, not directly comparable to the previous ones.
We shall investigate what assumptions are sufficient to yield sharp asy
mptotic estimates for the averages of $f$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Bell (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20201117T010000Z
DTEND;VALUE=DATE-TIME:20201117T020000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/66
DESCRIPTION:Title: A transcendental dynamical degree\nby Jason Bell (University of
Waterloo) as part of Number Theory Web Seminar\n\n\nAbstract\nThe degree o
f a dominant rational map $f:\\mathbb{P}^n\\to \\mathbb{P}^n$ is the commo
n degree of its homogeneous components. By considering iterates of $f$\,
one can form a sequence ${\\rm deg}(f^n)$\, which is submultiplicative and
hence has the property that there is some $\\lambda\\ge 1$ such that $({\
\rm deg}(f^n))^{1/n}\\to \\lambda$. The quantity $\\lambda$ is called the
first dynamical degree of $f$. We’ll give an overview of the significa
nce of the dynamical degree in complex dynamics and describe an example in
which this dynamical degree is provably transcendental. This is joint wo
rk with Jeffrey Diller and Mattias Jonsson.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Imre Ruzsa (Alfréd Rényi Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20210107T160000Z
DTEND;VALUE=DATE-TIME:20210107T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/67
DESCRIPTION:Title: Additive decomposition of signed primes\nby Imre Ruzsa (Alfréd
Rényi Institute of Mathematics) as part of Number Theory Web Seminar\n\n\
nAbstract\nAssuming the prime-tuple hypothesis\, the set of signed primes
is a sumset. More exactly\, there are infinite sets $A$\, $B$ of integers
such that $A+B$ consists exactly of the (positive and negative) primes wit
h $|p|>3$.\nI will also meditate on the possibility of a triple sum and an
alogous problems for the set of squarefree numbers.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (Institute for Advanced Study and Princeton Universit
y)
DTSTART;VALUE=DATE-TIME:20210114T160000Z
DTEND;VALUE=DATE-TIME:20210114T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/68
DESCRIPTION:Title: Summation formulae in spectral theory and number theory (A talk in h
onor of Zeev Rudnick's 60th Birthday)\nby Peter Sarnak (Institute for
Advanced Study and Princeton University) as part of Number Theory Web Semi
nar\n\n\nAbstract\nThe Poisson Summation formula\, Riemann-Guinand-Weil ex
plicit formula\, Selberg Trace Formula and Lefschetz Trace formula in the
function field\, are starting points for a number of Zeev Rudnick's works.
We will review some of these before describing some recent applications (
joint with P. Kurasov) of Lang's $\\mathbb{G}_m$ conjectures to the additi
ve structure of the spectra of metric graphs and crystalline measures.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary-Soroker (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20210121T160000Z
DTEND;VALUE=DATE-TIME:20210121T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/69
DESCRIPTION:Title: Random Polynomials\, Probabilistic Galois Theory\, and Finite Field
Arithmetic\nby Lior Bary-Soroker (Tel Aviv University) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nAbstract: In the talk we will discu
ss recent advances on the following two questions: \n\nLet $A(X) = \\sum \
\pm X^i$ be a random polynomial of degree $n$ with coefficients taking th
e values $-1\,1$ independently each with probability $1/2$.\n\nQ1: What is
the probability that $A$ is irreducible as the degree goes to infinity?\n
\nQ2: What is the typical Galois group of $A$?\n\nOne believes that the an
swers are YES and THE FULL SYMMETRIC GROUP\, respectively. These questions
were studied extensively in recent years\, and we will survey the tools
developed to attack these problems and partial results.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Banks (University of Missouri)
DTSTART;VALUE=DATE-TIME:20210128T160000Z
DTEND;VALUE=DATE-TIME:20210128T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/70
DESCRIPTION:Title: On the distribution of reduced fractions with squarefree denominator
s\nby William Banks (University of Missouri) as part of Number Theory
Web Seminar\n\n\nAbstract\nAbstract: In this talk we discuss how the nonva
nishing of the Riemann zeta function in a half-plane $\\{\\sigma>\\sigma_0
\\}$\, with some real $\\sigma_0<1$\, is equivalent to a strong statement
about the distribution in the unit interval of reduced fractions with squ
arefree denominators.\n\nThe approach utilizes an unconditional generaliza
tion of a theorem of Blomer concerning the distribution "on average" of s
quarefree integers in arithmetic progressions to large moduli.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (University of Bristol)
DTSTART;VALUE=DATE-TIME:20210204T160000Z
DTEND;VALUE=DATE-TIME:20210204T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/71
DESCRIPTION:Title: On the zeros of Fekete polynomials\nby Oleksiy Klurman (Universi
ty of Bristol) as part of Number Theory Web Seminar\n\n\nAbstract\nSince t
heir discovery by Dirichlet in the nineteenth century\, Fekete polynomials
(with coefficients being Legendre symbols) and their zeros attracted cons
iderable attention\, in particular\, due to their intimate connection with
putative Siegel zero and small class number problem. The goal of this tal
k is to discuss what we knew\, know and would like to know about zeros of
such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.
\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Don Zagier (Max Planck Institute for Mathematics)
DTSTART;VALUE=DATE-TIME:20210211T160000Z
DTEND;VALUE=DATE-TIME:20210211T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/72
DESCRIPTION:Title: Analytic functions related to zeta-values\, cotangent products\, and
the cohomology of $SL_2(\\Z)$\nby Don Zagier (Max Planck Institute fo
r Mathematics) as part of Number Theory Web Seminar\n\n\nAbstract\nI will
report on the properties of various functions\, going back essentially to
Herglotz\, that relate to a number of different topics in number theory\,
including those in the title but also others like Hecke operators or Stark
's conjectures. This is joint work with Danylo Radchenko.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Dill (University of Oxford)
DTSTART;VALUE=DATE-TIME:20210218T160000Z
DTEND;VALUE=DATE-TIME:20210218T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/73
DESCRIPTION:Title: Unlikely Intersections and Distinguished Categories\nby Gabriel
Dill (University of Oxford) as part of Number Theory Web Seminar\n\n\nAbst
ract\nAfter a general introduction to the field of unlikely intersections\
, I present current work in progress with Fabrizio Barroero\, in which we
propose an axiomatic approach towards studying unlikely intersections by i
ntroducing the framework of distinguished categories. This includes commut
ative algebraic groups and mixed Shimura varieties. It allows to us to def
ine all basic concepts of the field and prove some fundamental facts about
them\, e.g. the defect condition. In some categories that we call very di
stinguished\, we are able to show some implications between Zilber-Pink st
atements with respect to base change. This also yields new unconditional r
esults on the Zilber-Pink conjecture for curves in various contexts.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanju Velani (University of York)
DTSTART;VALUE=DATE-TIME:20210304T200000Z
DTEND;VALUE=DATE-TIME:20210304T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/74
DESCRIPTION:Title: The Shrinking Target Problem for Matrix Transformations of Tori\
nby Sanju Velani (University of York) as part of Number Theory Web Seminar
\n\n\nAbstract\nLet $T$ be a $d\\times d$ matrix with integral coefficient
s.\nThen $T$ determines a self-map of the $d$-dimensional torus $\\mathbb{
T}^d=\\R^d/\\Z^d$.\nChoose for each natural number $n$ a ball $B(n)$ in $X
$\n and suppose that $B(n+1)$ has smaller radius than $B(n)$ for all $n$.\
nThus the ball shrinks as $n$ increases. \nNow let $W$ be the set of point
s $x\\in \\mathbb{T}^d$ such that\n $T^n(x)\\in B(n)$ for infinitely many
$n\\in\n$. The size of $W$ measured in terms of $d$-dimensional Lebesgue m
easure (restricted to $\\mathbb{T}^d$) and Haudsorff dimension are pretty
much well understood. \n In this talk I explore the situation in which th
e points $ x \\in \\mathbb{T}^d$ are restricted to a nice subset ${\\mat
hcal M}$ (such as an analytic sub-manifold) of $\\mathbb{T}^d$\; that is\,
the points of interest are functionally dependent. I will essentially co
ncentrate on the situation when $d=2$\, $T$ has first row $(2\,0) $ and
second row $(0\,3)$\n and ${\\mathcal M}$ is the diagonal. In this specia
l case\, given a decreasing function $\\psi$\, understanding the shrink
ing target set $W \\cap {\\mathcal M}$ is equivalent to understanding the
set of $x\\in [0\,1]$ such that $ \\max\\{\\|2^nx\\|\, \\|3^nx\\|\\}<\\psi
(n) $ for infinitely many $n\\in\n$. \n \n\n \n This is joint work with B
ing Li (South China University of Technology)\, Lingmin Liao (UPEC) and Ev
geniy Zorin (York).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia University)
DTSTART;VALUE=DATE-TIME:20210311T200000Z
DTEND;VALUE=DATE-TIME:20210311T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/76
DESCRIPTION:Title: Moments and non-vanishing of cubic Dirichlet $L$-functions at $s=\\f
rac{1}{2}$\nby Chantal David (Concordia University) as part of Number
Theory Web Seminar\n\n\nAbstract\nA famous conjecture of Chowla predicts t
hat $L(\\frac{1}{2}\,\\chi)\\ne 0$ for all Dirichlet $L$-functions\nattach
ed to primitive characters $\\chi$. It was conjectured first in the case w
here $\\chi$ is a quadratic\ncharacter\, which is the most studied case. F
or quadratic Dirichlet $L$-functions\, Soundararajan\nproved that at least
87.5% of the quadratic Dirichlet $L$-functions do not vanish at $s=\\frac
{1}{2}$.\nUnder GRH\, there are slightly stronger results by Ozlek and Sny
der.\n\nWe present in this talk the first result showing a positive propor
tion of cubic Dirichlet\n$L$-functions non-vanishing at $s=\\frac{1}{2}$ f
or the non-Kummer case over function fields. This can\nbe achieved by usin
g the recent breakthrough work on sharp upper bounds for moments of\nSound
ararajan\, Harper and Lester-Radziwill. Our results would transfer over nu
mber fields\,\nbut we would need to assume GRH in this case.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shabnam Akhtari (University of Oregon)
DTSTART;VALUE=DATE-TIME:20210318T200000Z
DTEND;VALUE=DATE-TIME:20210318T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/77
DESCRIPTION:Title: Orders in Quartic Number Fields and Classical Diophantine Equations<
/a>\nby Shabnam Akhtari (University of Oregon) as part of Number Theory We
b Seminar\n\n\nAbstract\nAn order $\\mathcal{O}$ in an algebraic number fi
eld is called monogenic if over $\\mathbb{Z}$ it can be generated by one e
lement. Gy\\H{o}ry has shown that there are finitely equivalence classes \
n$\\alpha \\in \\mathcal{O}$ such that $\\mathcal{O} = \\mathbb{Z}[\\alpha
]$\, where two algebraic integers $\\alpha$ and $\\alpha'$ are called equi
valent if $\\alpha + \\alpha'$ or $\\alpha - \\alpha'$ is a rational inte
ger. An interesting problem is to count the number of monogenizations of
a given monogenic order. First we will note\, for a given order $\\mathcal
{O}$\, that \n$$\n\\mathcal{O} = \\mathbb{Z}[\\alpha] \\\, \\quad \\textrm
{in} \\\, \\\, \\alpha\,\n$$\nis indeed a Diophantine equation. Then we wi
ll modify some old algorithmic results to obtain new and improved upper bo
unds for the number of monogenizations of a quartic order.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitaly Bergelson (Ohio State University)
DTSTART;VALUE=DATE-TIME:20210325T203000Z
DTEND;VALUE=DATE-TIME:20210325T213000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/78
DESCRIPTION:Title: A "soft" dynamical approach to the Prime Number Theorem and disjoint
ness of additive and multiplicative semigroup actions\nby Vitaly Berge
lson (Ohio State University) as part of Number Theory Web Seminar\n\n\nAbs
tract\nWe will discuss a new type of ergodic theorem which has among its c
orollaries numerous classical results from multiplicative number theory\,
including the Prime Number Theorem\, a theorem of Pillai-Selberg and a the
orem of Erdős-Delange. This ergodic approach leads to a new dynamical fra
mework for a general form of Sarnak’s Möbius disjointness conjecture wh
ich focuses on the "joint independence" of actions of $(\n\,+)$ and $(\n\,
×)$. The talk is based on recent joint work with Florian Richter.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (Université Claude Bernard Lyon 1)
DTSTART;VALUE=DATE-TIME:20210401T150000Z
DTEND;VALUE=DATE-TIME:20210401T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/79
DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite automata
\nby Boris Adamczewski (Université Claude Bernard Lyon 1) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nIt is commonly expected that expansio
ns of numbers in multiplicatively independent bases\, such as 2 and 10\, s
hould have no common structure. However\, it seems extraordinarily difficu
lt to confirm this naive heuristic principle in some way or another. In th
e late 1960s\, Furstenberg suggested a series of conjectures\, which becam
e famous and aim to capture this heuristic. The work I will discuss in thi
s talk is motivated by one of these conjectures. Despite recent remarkable
progress by Shmerkin and Wu\, it remains totally out of reach of the curr
ent methods. While Furstenberg’s conjectures take place in a dynamical s
etting\, I will use instead the language of automata theory to formulate s
ome related problems that formalize and express in a different way the sam
e general heuristic. I will explain how the latter can be solved thanks to
some recent advances in Mahler’s method\; a method in transcendental nu
mber theory initiated by Mahler at the end of the 1920s. This a joint work
with Colin Faverjon.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:János Pintz (Alfréd Rényi Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20210408T150000Z
DTEND;VALUE=DATE-TIME:20210408T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/80
DESCRIPTION:Title: On the mean value of the remainder term of the prime number formula<
/a>\nby János Pintz (Alfréd Rényi Institute of Mathematics) as part of
Number Theory Web Seminar\n\n\nAbstract\nThere are several methods to obta
in a lower bound for the mean value of the absolute value of the remainder
term of the prime number formula as function of a hypothetical zero of th
e Riemann Zeta function off the critical line. (The case when the Riemann
Hypothesis is true can be treated easier.) The most efficient ones include
results of Knapowski-Turán\, Sz. Gy. Révész \, and the author\, proved
by several different methods\n\nThe result to be proved in the lecture pr
ovides (again with an other method) a quite good lower bound and it has th
e good feature (which is useful in further applications too) that instead
of the whole interval $[0\,X]$ it gives a good lower bound for the average
on $[F(X)\, X]$ with $\\log F(X)$ close to $\\log X$ (that is on "short"
intervals measured with the logarithmic scale).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Keating (University of Oxford)
DTSTART;VALUE=DATE-TIME:20210415T150000Z
DTEND;VALUE=DATE-TIME:20210415T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/81
DESCRIPTION:Title: Joint Moments\nby Jonathan Keating (University of Oxford) as par
t of Number Theory Web Seminar\n\n\nAbstract\nI will discuss the joint mom
ents of the Riemann zeta-function and its derivative\, and the correspondi
ng joint moments of the characteristic polynomials of random unitary matri
ces and their derivatives.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20210506T150000Z
DTEND;VALUE=DATE-TIME:20210506T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/82
DESCRIPTION:Title: A brief history of Hecke operators\nby Akshay Venkatesh (Institu
te for Advanced Study) as part of Number Theory Web Seminar\n\n\nAbstract\
nThis is an expository lecture about Hecke operators\, in the context of n
umber theory. We will trace some of the history of the ideas\, starting b
efore Hecke's birth and proceeding through the subsequent century. In part
icular we will discuss some of the original motivations and then the impac
t of ideas from representation theory and algebraic geometry. This lecture
is aimed at non-experts.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Kontorovich (Rutgers University)
DTSTART;VALUE=DATE-TIME:20210513T150000Z
DTEND;VALUE=DATE-TIME:20210513T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/83
DESCRIPTION:Title: Arithmetic Groups and Sphere Packings\nby Alex Kontorovich (Rutg
ers University) as part of Number Theory Web Seminar\n\n\nAbstract\nWe dis
cuss recent progress on understanding connections between the objects in t
he title.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pietro Corvaja (University of Udine)
DTSTART;VALUE=DATE-TIME:20210429T150000Z
DTEND;VALUE=DATE-TIME:20210429T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/84
DESCRIPTION:Title: On the local-to-global principle for value sets\nby Pietro Corva
ja (University of Udine) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nGiven a finite morphism $f: X \\to Y$ between algebraic curves over num
ber fields\, we study the set of rational (or integral) points in $Y$ havi
ng a pre-image in every $p$-adic completion of the number field\, but no r
ational pre-images. In particular\, we investigate whether this set can be
infinite.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Tichy (TU Graz)
DTSTART;VALUE=DATE-TIME:20210527T150000Z
DTEND;VALUE=DATE-TIME:20210527T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/85
DESCRIPTION:Title: Equidistribution\, exponential sums and van der Corput sets\nby
Robert Tichy (TU Graz) as part of Number Theory Web Seminar\n\n\nAbstract\
nThe talk starts with a survey on Sarkoezy`s results on difference sets an
d with Furstenberg`s dynamic approach to additive problems. We present som
e results of a joint work with Bergelson\, Kolesnik\, Son and Madritsch co
ncerning multidimensional van der Corput sets based on new bounds for expo
nential sums. In a second part we give a brief introduction on equidistrib
ution theory focusing on the interplay of exponential sums with difference
theorems. In a third part Hardy fields are discussed in some detail. This
concept was introduced to equidistribution theory by Boshernitzan and it
tuned out to be very fruitful. We will report on recent results of Bergel
son et al. and at the very end on applications to diophantine approximatio
n. This includes results concerning the approximation of polynomial-like f
unctions along primes which were established in a joint work with Madritsc
h and sharpened very recently by my PhD student Minelli.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renate Scheidler (University of Calgary)
DTSTART;VALUE=DATE-TIME:20210422T150000Z
DTEND;VALUE=DATE-TIME:20210422T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/86
DESCRIPTION:Title: Computing modular polynomials and isogeny graphs of rank $2$ Drinfel
d modules\nby Renate Scheidler (University of Calgary) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nDrinfeld modules represent the functi
on field analogue of the theory of complex multiplication. They were intro
duced as "elliptic modules" by Vladimir Drinfeld in the 1970s in the cours
e of proving the Langlands conjectures for $\\GL(2)$ over global function
fields. Drinfeld modules of rank $2$ exhibit very similar behaviour to ell
iptic curves: they are classified as ordinary or supersingular\, support i
sogenies and their duals\, and their endomorphism rings have an analogous
structure. Their isomorphism classes are parameterized by $j$-invariants\,
and Drinfeld modular polynomials can be used to compute their isogeny gra
phs whose ordinary connected components take the shape of volcanos. While
the rich analytic and algebraic theory of Drinfeld modules has undergone e
xtensive investigation\, very little has been explored from a computationa
l perspective. This research represents the first foray in this direction\
, introducing an algorithm for computing Drinfeld modular polynomials and
isogeny graphs. \n\nThis is joint work with Perlas Caranay and Matt Greenb
erg\, as well as ongoing research with Edgar Pacheco Castan. Some familiar
ity with elliptic curves is expected for this talk\, but no prior knowledg
e of Drinfeld modules is assumed.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shou-Wu Zhang (Princeton University)
DTSTART;VALUE=DATE-TIME:20210617T150000Z
DTEND;VALUE=DATE-TIME:20210617T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/87
DESCRIPTION:Title: Adelic line bundles over quasi-projective varieties\nby Shou-Wu
Zhang (Princeton University) as part of Number Theory Web Seminar\n\n\nAbs
tract\nFor quasi-projective varieties over finitely generated fields\, we
develop a theory of adelic line bundles including an equidistribution theo
rem for Galois orbits of small points. In this lecture\, we will explain t
his theory and its application to arithmetic of abelian varieties\, dynami
cal systems\, and their moduli. This is a joint work with Xinyi Yuan.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Chambert-Loir (Université Paris-Diderot)
DTSTART;VALUE=DATE-TIME:20210603T180000Z
DTEND;VALUE=DATE-TIME:20210603T190000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/88
DESCRIPTION:Title: From complex function theory to non-archimedean spaces - a number th
eoretical thread\nby Antoine Chambert-Loir (Université Paris-Diderot)
as part of Number Theory Web Seminar\n\n\nAbstract\nDiophantine geometry
and complex function theory have a long and well known history of mutual f
riendship\, attested\, for example\, by the fruitful interactions between
height functions and potential theory. In the last 50 years\, interactions
even deepened with the invention of Arakelov geometry (Arakelov\, Gillet/
Soulé\, Faltings) and its application by Szpiro/Ullmo/Zhang to equidistri
bution theorems and the Bogomolov conjecture. Roughly at the same time\, B
erkovich invented a new kind of non-archimedean analytic spaces which poss
ess a rich\nand well behaved geometric structure. This opened the way to n
on-archimedean potential theory (Baker/Rumely\, Favre/Rivera-Letelier)\, o
r to arithmetic/geometric equidistribution theorems in this case. More rec
ently\, Ducros and myself introduced basic ideas from tropical geometry an
d a construction of Lagerberg to construct a calculus of $(p\,q)$-forms on
Berkovich spaces\, which is an analogue of the corresponding calculus on
complex manifolds\, and seems to be an attractive candidate for being the
$p$-adic side of height function theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Silverberg (University of California\, Irvine)
DTSTART;VALUE=DATE-TIME:20210520T150000Z
DTEND;VALUE=DATE-TIME:20210520T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/89
DESCRIPTION:Title: Cryptographic Multilinear Maps and Miscellaneous Musings\nby Ali
ce Silverberg (University of California\, Irvine) as part of Number Theory
Web Seminar\n\n\nAbstract\nRecognizing that many of us have Zoom fatigue\
, I will keep this talk light\, without too many technical details. In add
ition to discussing an open problem on multilinear maps that has applicati
ons to cryptography\, I'll give miscellaneous musings about things I've le
arned over the years that I wish I'd learned sooner.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annette Huber-Klawitter (University of Freiburg)
DTSTART;VALUE=DATE-TIME:20210624T150000Z
DTEND;VALUE=DATE-TIME:20210624T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/90
DESCRIPTION:Title: Periods and O-minimality\nby Annette Huber-Klawitter (University
of Freiburg) as part of Number Theory Web Seminar\n\n\nAbstract\nRoughly\
, periods are numbers obtained by integrating algebraic\ndifferential form
s over domains of integration also of arithmetic\nnature. I am going to g
ive a survey on the state of the period\nconjecture and different points o
f view. I also want to present a\nrelation to o-minimal geometry.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Young (Texas A&M University)
DTSTART;VALUE=DATE-TIME:20210610T150000Z
DTEND;VALUE=DATE-TIME:20210610T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/91
DESCRIPTION:Title: The Weyl bound for Dirichlet L-functions\nby Matthew Young (Texa
s A&M University) as part of Number Theory Web Seminar\n\n\nAbstract\nTher
e is an analogy between the behavior of the Riemann zeta function high in
the critical strip\, and the behavior of Dirichlet $L$-functions of large
conductors. In many important problems\, our understanding of Dirichlet $
L$-functions is weaker than for zeta\; for example\, the zero-free regions
are not of the same quality due to the possible Landau-Siegel zero. This
talk will discuss recent progress (joint with Ian Petrow) on subconvexity
bounds for Dirichlet $L$-functions. These new bounds now match the origin
al subconvexity bound for the zeta function derived by Hardy and Littlewoo
d using Weyl's differencing method.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Green
DTSTART;VALUE=DATE-TIME:20210225T160000Z
DTEND;VALUE=DATE-TIME:20210225T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/92
DESCRIPTION:Title: New lower bounds for van der Waerden numbers\nby Ben Green as pa
rt of Number Theory Web Seminar\n\n\nAbstract\nColour $\\{1\,..\,N\\}$ red
and blue\, in such a manner that no $3$ of the blue elements are in arith
metic progression. How long an arithmetic progression of red elements must
there be? It had been speculated based on numerical evidence that there m
ust always be a red progression of length about $\\sqrt{N}$. I will descri
be a construction which shows that this is not the case - in fact\, there
is a colouring with no red progression of length more than about $\\exp ((
\\log N)^{3/4})$\, and in particular less than any fixed power of $N$.\n\n
I will give a general overview of this kind of problem (which can be formu
lated in terms of finding lower bounds for so-called van der Waerden numbe
rs)\, and an overview of the construction and some of the ingredients whic
h enter into the proof. The collection of techniques brought to bear on th
e problem is quite extensive and includes tools from diophantine approxima
tion\, additive number theory and\, at one point\, random matrix theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjul Bhargava (Princeton University)
DTSTART;VALUE=DATE-TIME:20210701T150000Z
DTEND;VALUE=DATE-TIME:20210701T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/93
DESCRIPTION:Title: Galois groups of random integer polynomials (A talk in honor of Don
Zagier's 70th birthday)\nby Manjul Bhargava (Princeton University) as
part of Number Theory Web Seminar\n\n\nAbstract\nOf the $(2H+1)^n$ monic i
nteger polynomials $f(x)=x^n+a_1 x^{n-1}+\\cdots+a_n$ with $\\max\\{|a_1|\
,\\ldots\,|a_n|\\}\\leq H$\, how many have associated Galois group that is
not the full symmetric group $S_n$? There are clearly $\\gg H^{n-1}$ such
polynomials\, as can be seen by setting $a_n=0$. In 1936\, van der Waerde
n conjectured that $O(H^{n-1})$ should in fact also be the correct upper b
ound for the count of such polynomials. The conjecture has been known for
$n\\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this
talk\, we prove the "Weak van der Waerden Conjecture"\, which states that
the number of such polynomials is $O_\\epsilon(H^{n-1+\\epsilon})$\, for a
ll degrees $n$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Conrey (American Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20210708T150000Z
DTEND;VALUE=DATE-TIME:20210708T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/94
DESCRIPTION:Title: Moments\, ratios\, arithmetic functions in short intervals and rando
m matrix averages\nby Brian Conrey (American Institute of Mathematics)
as part of Number Theory Web Seminar\n\n\nAbstract\nWe discuss how the co
njectures for moments of $L$-functions\nimply short interval averages of t
he $L$-coefficient convolutions. Similarly\nthe ratios conjectures lead to
short interval averages of the convolutions\nof coefficients at almost pr
imes. These in turn are related to random matrix averages considered by Di
aconis - Gamburd and by Diaconis - Shahshahani.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ricardo Menares (Pontificia Universidad Católica de Chile)
DTSTART;VALUE=DATE-TIME:20210715T150000Z
DTEND;VALUE=DATE-TIME:20210715T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/95
DESCRIPTION:Title: $p$-adic distribution of CM points\nby Ricardo Menares (Pontific
ia Universidad Católica de Chile) as part of Number Theory Web Seminar\n\
n\nAbstract\nCM points are the isomorphism classes of CM elliptic curves.
When ordered by the absolute value of the discriminant of the endomorphism
ring\, CM points are distributed along the complex (level one) modular cu
rve according to the hyperbolic measure. This statement was proved by Duke
for fundamental discriminants and later\, building on this work\, Clozel
and Ullmo proved it in full generality.\n\nIn this talk\, we establish the
$p$-adic analogue of this result. Namely\, for a fixed prime $p$ we regar
d the CM points as a subset of the $p$-adic space attached to the modular
curve and we classify the possible accumulation measures of CM points as t
he discriminant varies. In particular\, we find that there are infinitely
many such measures. This is in stark contrast to the complex case\, where
the hyperbolic measure is the unique accumulation measure. \n\nAs an appli
cation\, we show that for any finite set $S$ of prime numbers\, the set of
singular moduli which are $S$-units is finite.\n\nThis is joint work with
Sebastián Herrero (PUC Valparaíso) and Juan Rivera-Letelier (Rochester)
.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Kühne (University of Copenhagen)
DTSTART;VALUE=DATE-TIME:20210902T150000Z
DTEND;VALUE=DATE-TIME:20210902T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/96
DESCRIPTION:Title: The uniform Bogomolov conjecture for algebraic curves\nby Lars K
ühne (University of Copenhagen) as part of Number Theory Web Seminar\n\n\
nAbstract\nI will present an equidistribution result for families of (non-
degenerate) subvarieties in a (general) family of abelian varieties. This
extends a result of DeMarco and Mavraki for curves in fibered products of
elliptic surfaces\, but it also follows from independent work by Yuan and
Zhang\, which has been recently reported in this seminar. I will therefore
focus on the application that motivated my work\, namely a uniform versio
n of the classical Bogomolov conjecture for curves embedded in their Jacob
ians. This has been previously only known in a few select cases by work of
David–Philippon and DeMarco–Krieger–Ye. Furthermore\, one can deduc
e a rather uniform version of the Mordell-Lang conjecture by complementing
a result of Dimitrov–Gao–Habegger: The number of rational points on a
smooth algebraic curve defined over a number field can be bounded solely
in terms of its genus and the Mordell-Weil rank of its Jacobian. Again\, t
his was previously known only under additional assumptions (Stoll\, Katz
–Rabinoff–Zureick-Brown). All these results have been recently general
ized beyond curves in joint work with Ziyang Gao and Tangli Ge\, but I wil
l restrict to the case of curves for simplicity.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Marc Deshouillers (Institut de Mathématiques de Bordeaux)
DTSTART;VALUE=DATE-TIME:20211014T150000Z
DTEND;VALUE=DATE-TIME:20211014T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/97
DESCRIPTION:Title: Are factorials sums of three cubes?\nby Jean-Marc Deshouillers (
Institut de Mathématiques de Bordeaux) as part of Number Theory Web Semin
ar\n\n\nAbstract\nLet $\\mathcal{C}_3$ be the set of non-negative integer
s which are sums of the cubes of three non-negative integers and let $C_3
$ be their counting function\, id est\n$$\nC_3(x)= \\operatorname{Card}\\{
n \\le x \\colon n \\in \\mathcal{C}_3\\}.\n$$\nOur knowledge of sums of t
hree cubes is somewhat limited\, for example\, we do not know whether ther
e exists a positive real $c$ such that for any sufficiently large $x$ one
has\n$$\nC_3(x) \\ge cx.\n$$\nNumerical and probabilistic results are in f
avour of \n$$\nC_3(x) \\sim cx\, \\text{ where } c=0.0999425... \\text{ as
$x$ tends to infinity}.\n$$\n\nNumerical results presented in the chapter
A267414 of the OEIS project suggest that factorials are very often sums
of three cubes and even that as soon as $n$ is large enough\, $n!$ is a su
m of three cubes. The aim of the talk is to present a probability model\,
consistent with the actual distribution of cubes\, in which\, almost sure
ly\, as soon as $n$ is large enough\, $n!$ is a sum of three pseudo-cubes.
\n\nWe shall also give two applications of our result to classical problem
s on sums of cubes. \nThe result presented in the talk have been jointly
obtained with Altug Alkan (Istanbul)\, François Hennecart (Saint Étienne
) et Bernard Landreau (Angers).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Sombra (ICREA and University of Barcelona)
DTSTART;VALUE=DATE-TIME:20210916T150000Z
DTEND;VALUE=DATE-TIME:20210916T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/98
DESCRIPTION:Title: The mean height of the solution set of a system of polynomial equati
ons\nby Martin Sombra (ICREA and University of Barcelona) as part of N
umber Theory Web Seminar\n\n\nAbstract\nBernstein’s theorem allows to pr
edict the number of solutions of a system of Laurent\npolynomial equations
in terms of combinatorial invariants. When the coefficients of the system
\nare algebraic numbers\, we can ask about the height of these solutions.
Based on an on-going project with Roberto Gualdi (Regensburg)\, I will exp
lain how one can approach this question using tools from the Arakelov geom
etry of toric varieties.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos (University of Montreal)
DTSTART;VALUE=DATE-TIME:20211028T150000Z
DTEND;VALUE=DATE-TIME:20211028T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/99
DESCRIPTION:Title: Towards a high-dimensional theory of divisors of integers\nby Di
mitris Koukoulopoulos (University of Montreal) as part of Number Theory We
b Seminar\n\n\nAbstract\nIn this talk\, I will survey some results about h
igh-dimensional phenomena in the theory of divisors of integers. \n\nFix a
n integer $k\\ge2$ and pick an integer $n\\le x$ uniformly at random. We t
hen consider the following two basic problems:\nWhat are the chances that
$n$ can be factored as $n=d_1\\cdots d_k$ with each factor $d_i$ lying in
some prescribed dyadic interval $[y_i\,2y_i]$?\nWhat are the chances that
we can find $k$ divisors of $n$\, say $d_1\,\\dots\,d_k$\, such that $|\\l
og(d_j/d_i)|<1$ for all $i\,j$\, and which are all composed from a prescri
bed set of prime factors of $n$?\nThe first problem is a high-dimensional
generalization of the Erdős multiplication table problem\; it is well-und
erstood when $k\\le 6$\, but less so when $k\\ge7$. The second problem is
related to Hooley’s function $\\Delta(n):=\\max_u \\#\\{d|n:u<\\log d\\l
e u+1\\}$ that measures the concentration of the sequence of divisors of $
n$\, and that has surprising applications to Diophantine number theory.\n\
nIn recent work with Kevin Ford and Ben Green\, we built on the earlier wo
rk on Problem 1 to develop a new approach to Problem 2. This led to an imp
roved lower bound on the almost-sure behaviour of Hooley’s $\\Delta$-fun
ction\, that we conjecture to be optimal. The new ideas might in turn shed
light to Problem 1 and other high-dimensional phenomena about divisors of
integers.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Fehm (Technische Universität Dresden)
DTSTART;VALUE=DATE-TIME:20210729T150000Z
DTEND;VALUE=DATE-TIME:20210729T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/100
DESCRIPTION:Title: Is $\\Z$ diophantine in $\\Q$?\nby Arno Fehm (Technische Univer
sität Dresden) as part of Number Theory Web Seminar\n\n\nAbstract\nAre th
e integers the projection of the rational zeros of a polynomial in several
variables onto the first coordinate? The aim of this talk is to motivate
and discuss this longstanding question. I will survey some results regardi
ng diophantine sets and Hilbert's tenth problem (the existence of an algor
ithm that decides whether a polynomial has a zero) in fields and will disc
uss a few conjectures\, some classical and some more recent\, that suggest
that the answer to the question should be negative.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandru Zaharescu (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20210826T210000Z
DTEND;VALUE=DATE-TIME:20210826T220000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/101
DESCRIPTION:Title: Some remarks on Landau - Siegel zeros\nby Alexandru Zaharescu (
University of Illinois at Urbana-Champaign) as part of Number Theory Web S
eminar\n\n\nAbstract\nIn the first part of the talk I will survey some kno
wn results related to the hypothetical existence of Landau - Siegel zeros.
In the second part of the talk I will discuss some recent joint work with
Hung Bui and Kyle Pratt in which we show that the existence of Landau - S
iegel zeros has implications for the behavior of $L$ - functions at the ce
ntral point.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210805T150000Z
DTEND;VALUE=DATE-TIME:20210805T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/102
DESCRIPTION:Title: Digits\nby Frank Calegari (University of Chicago) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nWe discuss some results concerning
the decimal expansion of $1/p$ for primes $p$\, some due to Gauss\, and so
me from the present day. This is work in progress with Soundararajan which
we may well write up one day.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kumar Murty (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210722T150000Z
DTEND;VALUE=DATE-TIME:20210722T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/103
DESCRIPTION:Title: Periods and Mixed Motives\nby Kumar Murty (University of Toront
o) as part of Number Theory Web Seminar\n\n\nAbstract\nWe discuss some con
sequences of Grothendieck's Period Conjecture in the context of mixed moti
ves. In particular\, this conjecture implies that $\\zeta(3)$\, $\\log 2$
and $\\pi$ are algebraically independent (contrary to an expectation of Eu
ler). After some 'motivation' and introductory remarks on periods\, we der
ive our consequences as a result of studying mixed motives whose Galois gr
oup has a large unipotent radical. This is joint work with Payman Eskandar
i.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Amoroso (University of Caen)
DTSTART;VALUE=DATE-TIME:20210812T150000Z
DTEND;VALUE=DATE-TIME:20210812T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/104
DESCRIPTION:Title: Bounded Height in Pencils of Subgroups of finite rank\nby Franc
esco Amoroso (University of Caen) as part of Number Theory Web Seminar\n\n
\nAbstract\n[Joint work with D. Masser and U. Zannier] \n\nLet $n>1$ be a
varying natural number. By a result of Beukers\, the solutions of $t^n+(1-
t)^n=1$ have uniformly bounded height. What happens if we allow rational e
xponents? \n\nWe consider the analogous question replacing the affine curv
e $x+y=1$ with an arbitrary irreducible curve and $\\{t^n | n \\textrm{ ra
tional}\\}$ with the division group of a finitely generated subgroup.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Kowalski (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20210909T150000Z
DTEND;VALUE=DATE-TIME:20210909T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/105
DESCRIPTION:Title: Harmonic analysis over finite fields and equidistribution\nby E
mmanuel Kowalski (ETH Zürich) as part of Number Theory Web Seminar\n\n\nA
bstract\nIn 1976\, Deligne defined a geometric version of the Fourier tran
sform over finite fields\, leading to significant applications in number t
heory.\n\nFor a number of applications\, including equidistribution of exp
onential sums parameterized by multiplicative characters\, it would be ver
y helpful to have a similar geometric harmonic analysis for other groups.
I will discuss ongoing joint work with A. Forey and J. Fresán in which we
establish some results in this direction by generalizing ideas of Katz. I
will present the general equidistribution theorem for exponential sums pa
rameterized by characters that we obtain\, and discuss applications\, as w
ell as challenges\, open questions and mysteries.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anish Ghosh (Tata Institute of Fundamental Research)
DTSTART;VALUE=DATE-TIME:20210930T150000Z
DTEND;VALUE=DATE-TIME:20210930T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/106
DESCRIPTION:Title: Values of quadratic forms at integer points\nby Anish Ghosh (Ta
ta Institute of Fundamental Research) as part of Number Theory Web Seminar
\n\n\nAbstract\nA famous theorem of Margulis\, resolving a conjecture of O
ppenheim\, states that an indefinite\, irrational quadratic form in at lea
st three variables takes a dense set of values at integer points. Recently
there has been a push towards establishing effective versions of Margulis
's theorem. I will explain Margulis's approach to this problem which invol
ves the ergodic theory of group actions on homogeneous spaces. I will then
discuss some new effective results in this direction. These results use a
variety of techniques including tools from ergodic theory\, analytic numb
er theory as well as the geometry of numbers.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Carmen Cojocaru (University of Illinois at Chicago and Insti
tute of Mathematics of the Romanian Academy)
DTSTART;VALUE=DATE-TIME:20210923T150000Z
DTEND;VALUE=DATE-TIME:20210923T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/107
DESCRIPTION:Title: Bounds for the distribution of the Frobenius traces associated to a
belian varieties\nby Alina Carmen Cojocaru (University of Illinois at
Chicago and Institute of Mathematics of the Romanian Academy) as part of N
umber Theory Web Seminar\n\n\nAbstract\nIn 1976\, Serge Lang and Hale Trot
ter conjectured the asymptotic growth of the number $\\pi_A(x\, t)$ of pri
mes $p < x$ for which the Frobenius trace $a_p$ of a non-CM elliptic curve
$A/\\mathbb{Q}$ equals an integer $t$. Even though their conjecture remai
ns open\, over the past decades the study of the counting function $\\pi_A
(x\, t)$ has witnessed remarkable advances. We will discuss generalization
s of such studies in the setting of an abelian variety $A/\\mathbb{Q}$ of
arbitrary dimension and we will present non-trivial upper bounds for the c
orresponding counting function $\\pi_A(x\, t)$. This is joint work with Ti
an Wang (University of Illinois at Chicago).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henryk Iwaniec (Rutgers University)
DTSTART;VALUE=DATE-TIME:20211007T150000Z
DTEND;VALUE=DATE-TIME:20211007T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/108
DESCRIPTION:Title: Remarks on the large sieve (A talk in honor of John Friedlander's 8
0th birthday)\nby Henryk Iwaniec (Rutgers University) as part of Numbe
r Theory Web Seminar\n\n\nAbstract\nThe concept of the large sieve will be
discussed in various contexts. The power and limitation of basic estimate
s will be illustrated with some examples. Recent work on the large sieve f
or characters to prime moduli will be explained.\n\nSpecial Chairs: Leo Go
ldmakher (Williams College) and Andrew Granville (University of Montreal)\
n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard University)
DTSTART;VALUE=DATE-TIME:20211118T160000Z
DTEND;VALUE=DATE-TIME:20211118T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/109
DESCRIPTION:Title: Towards uniformity in the dynamical Bogomolov conjecture\nby My
rto Mavraki (Harvard University) as part of Number Theory Web Seminar\n\n\
nAbstract\nInspired by an analogy between torsion and preperiodic points\,
Zhang has proposed a dynamical generalization of the classical Manin-Mumf
ord and Bogomolov conjectures. A special case of these conjectures\, for `
split' maps\, has recently been established by Nguyen\, Ghioca and Ye. In
particular\, they show that two rational maps have at most finitely many c
ommon preperiodic points\, unless they are `related'. Recent breakthroughs
by Dimitrov\, Gao\, Habegger and Kühne have established that the classic
al Bogomolov conjecture holds uniformly across curves of given genus. \n\n
In this talk we discuss uniform versions of the dynamical Bogomolov conjec
ture across 1-parameter families of certain split maps. To this end\, we e
stablish an instance of a 'relative dynamical Bogomolov'. This is work in
progress joint with Harry Schmidt (University of Basel).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johan Commelin (Albert–Ludwigs-Universität Freiburg)
DTSTART;VALUE=DATE-TIME:20211021T150000Z
DTEND;VALUE=DATE-TIME:20211021T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/110
DESCRIPTION:Title: Liquid Tensor Experiment\nby Johan Commelin (Albert–Ludwigs-U
niversität Freiburg) as part of Number Theory Web Seminar\n\n\nAbstract\n
In December 2020\, Peter Scholze posed a challenge to formally verify the
main theorem on liquid $\\mathbb{R}$-vector spaces\, which is part of his
joint work with Dustin Clausen on condensed mathematics. I took up this ch
allenge with a team of mathematicians to verify the theorem in the Lean pr
oof assistant. Half a year later\, we reached a major milestone\, and our
expectation is that in a couple of months we will have completed the full
challenge.\n\nIn this talk I will give a brief motivation for condensed/li
quid mathematics\, a demonstration of the Lean proof assistant\, and discu
ss our experiences formalizing state-of-the-art research in mathematics.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Dvir (Princeton University)
DTSTART;VALUE=DATE-TIME:20210819T150000Z
DTEND;VALUE=DATE-TIME:20210819T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/111
DESCRIPTION:Title: The Kakeya set conjecture over rings of integers modulo square free
$m$\nby Zeev Dvir (Princeton University) as part of Number Theory Web
Seminar\n\n\nAbstract\nWe show that\, when $N$ is any square-free integer
\, the size of the smallest Kakeya set in $(ℤ/Nℤ)^n$ is at least $C_{\
\epsilon\,n}N^{n-\\epsilon}$ for any $\\epsilon>0$ -- resolving a special
case of a conjecture of Hickman and Wright. Previously\, such bounds were
only known for the case of prime $N$. We also show that the case of genera
l $N$ can be reduced to lower bounding the $p$-rank of the incidence matri
x of points and hyperplanes over $(ℤ/p^kℤ)^n$. Joint work with Manik D
har.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Skorobogatov (Imperial College London)
DTSTART;VALUE=DATE-TIME:20211125T160000Z
DTEND;VALUE=DATE-TIME:20211125T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/112
DESCRIPTION:Title: On uniformity conjectures for abelian varieties and K3 surfaces
\nby Alexei Skorobogatov (Imperial College London) as part of Number Theor
y Web Seminar\n\n\nAbstract\nI will discuss logical links among uniformity
conjectures concerning K3 surfaces and abelian varieties of bounded dimen
sion defined over number fields of bounded degree. The conjectures concern
the endomorphism algebra of an abelian variety\, the Néron–Severi latt
ice of a K3 surface\, and the Galois invariant subgroup of the geometric B
rauer group. The talk is based on a joint work with Martin Orr and Yuri Za
rhin.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katherine Stange (University of Colorado\, Boulder)
DTSTART;VALUE=DATE-TIME:20211104T160000Z
DTEND;VALUE=DATE-TIME:20211104T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/113
DESCRIPTION:Title: Algebraic Number Starscapes\nby Katherine Stange (University of
Colorado\, Boulder) as part of Number Theory Web Seminar\n\n\nAbstract\nI
n the spirit of experimentation\, at the Fall 2019 ICERM special semester
on “Illustrating Mathematics\,” I began drawing algebraic numbers in t
he complex plane. Edmund Harriss\, Steve Trettel and I sized the numbers
by arithmetic complexity and found a wealth of pattern and structure. In
this talk\, I’ll take you on a visual tour and share some of the mathema
tical explanations we found for what can be quite stunning pictures (in th
e hands of a mathematician and artist like Edmund). This experience gave
me a new perspective on complex Diophantine approximation: one can view a
pproximation properties as being dictated by the geometry of the map from
coefficient space to root space in different polynomial degrees. I’ll e
xplain this geometry\, and discuss a few Diophantine results\, known and n
ew\, in this context.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiran Kedlaya (University of California San Diego)
DTSTART;VALUE=DATE-TIME:20211202T160000Z
DTEND;VALUE=DATE-TIME:20211202T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/114
DESCRIPTION:Title: Orders of abelian varieties over $\\mathbb{F}_2$\nby Kiran Kedl
aya (University of California San Diego) as part of Number Theory Web Semi
nar\n\n\nAbstract\nWe describe several recent results on orders of abelian
varieties over $\\mathbb{F}_2$: every positive integer occurs as the orde
r of an ordinary abelian variety over $\\mathbb{F}_2$ (joint with E. Howe)
\; every positive integer occurs infinitely often as the order of a simple
abelian variety over $\\mathbb{F}_2$\; the geometric decomposition of the
simple abelian varieties over $\\mathbb{F}_2$ can be described explicitly
(joint with T. D'Nelly-Warady)\; and the relative class number one proble
m for function fields is reduced to a finite computation (work in progress
). All of these results rely on the relationship between isogeny classes o
f abelian varieties over finite fields and Weil polynomials given by the w
ork of Weil and Honda-Tate. With these results in hand\, most of the work
is to construct algebraic integers satisfying suitable archimedean constra
ints.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avi Wigderson (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20211111T160000Z
DTEND;VALUE=DATE-TIME:20211111T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/115
DESCRIPTION:Title: Randomness\nby Avi Wigderson (Institute for Advanced Study) as
part of Number Theory Web Seminar\n\n\nAbstract\nIs the universe inherentl
y deterministic or probabilistic? Perhaps more importantly - can we tell t
he difference between the two? \n\nHumanity has pondered the meaning and u
tility of randomness for millennia. \nThere is a remarkable variety of way
s in which we utilize perfect coin tosses to our advantage: in statistics\
, cryptography\, game theory\, algorithms\, gambling... Indeed\, randomnes
s seems indispensable! Which of these applications survive if the universe
had no (accessible) randomness in it at all? Which of them survive if onl
y poor quality randomness is available\, e.g. that arises from somewhat "u
npredictable" phenomena like the weather or the stock market? \n\nA comput
ational theory of randomness\, developed in the past several decades\, rev
eals (perhaps counter-intuitively) that very little is lost in such determ
inistic or weakly random worlds. In the talk I'll explain the main ideas a
nd results of this theory\, notions of pseudo-randomness\, and connections
to computational intractability. \n\nIt is interesting that Number Theory
played an important role throughout this development. It supplied problem
s whose algorithmic solution make randomness seem powerful\, problems for
which randomness can be eliminated from such solutions\, and problems wher
e the power of randomness remains a major challenge for computational comp
lexity theorists and mathematicians. I will use these problems (and others
) to demonstrate aspects of this theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London\, UK)
DTSTART;VALUE=DATE-TIME:20211216T160000Z
DTEND;VALUE=DATE-TIME:20211216T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/116
DESCRIPTION:Title: Euler systems and the Birch—Swinnerton-Dyer conjecture for abelia
n surfaces\nby Sarah Zerbes (University College London\, UK) as part o
f Number Theory Web Seminar\n\n\nAbstract\nEuler systems are one of the mo
st powerful tools for proving cases of the Bloch--Kato conjecture\, and ot
her related problems such as the Birch and Swinnerton-Dyer conjecture. \n\
nI will recall a series of recent works (variously joint with Loeffler\, P
illoni\, Skinner) giving rise to an Euler system in the cohomology of Shim
ura varieties for $\\mathrm{GSp}(4)$\, and an explicit reciprocity law rel
ating the Euler system to values of $L$-functions. I will then recent work
with Loeffler\, in which we use this Euler system to prove new cases of t
he BSD conjecture for modular abelian surfaces over $\\Q$\, and modular el
liptic curves over imaginary quadratic fields.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samir Siksek (University of Warwick)
DTSTART;VALUE=DATE-TIME:20211209T160000Z
DTEND;VALUE=DATE-TIME:20211209T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/117
DESCRIPTION:Title: The Fermat equation and the unit equation\nby Samir Siksek (Uni
versity of Warwick) as part of Number Theory Web Seminar\n\n\nAbstract\nTh
e asymptotic Fermat conjecture (AFC) states that for a number field $K$\,
and for sufficiently large primes $p$\, the only solutions to the Fermat e
quation $X^p+Y^p+Z^p=0$ in $K$ are the obvious ones. We sketch recent work
that connects the Fermat equation to the far more elementary unit equatio
n\, and explain how this surprising connection can be exploited to prove A
FC for several infinite families of number fields. This talk is based on j
oint work with Nuno Freitas\, Alain Kraus and Haluk Sengun.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Péter Varjú (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20220113T160000Z
DTEND;VALUE=DATE-TIME:20220113T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/118
DESCRIPTION:Title: Irreducibility of random polynomials\nby Péter Varjú (Univers
ity of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstract\nCons
ider random polynomials of degree $d$ whose leading and constant coefficie
nts are $1$ and the rest are independent taking the values $0$ or $1$ with
equal probability. A conjecture of Odlyzko and Poonen predicts that such
a polynomial is irreducible in $\\Z[x]$ with high probability as $d$ grow
s. This conjecture is still open\, but Emmanuel Breuillard and I proved it
assuming the Extended Riemann Hypothesis. I will briefly recall the metho
d of proof of this result and will discuss later developments that apply t
his method to other models of random polynomials.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ekin Özman (Boğaziçi University)
DTSTART;VALUE=DATE-TIME:20220303T160000Z
DTEND;VALUE=DATE-TIME:20220303T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/119
DESCRIPTION:Title: Modular Curves and Asymptotic Solutions to Fermat-type Equations\nby Ekin Özman (Boğaziçi University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nUnderstanding solutions of Diophantine equations over
rationals or more generally over any number field is one of the main probl
ems of number theory. By the help of the modular techniques used in the pr
oof of Fermat’s last theorem by Wiles and its generalizations\, it is po
ssible to solve other Diophantine equations too. Understanding quadratic p
oints on the classical modular curve play a central role in this approach.
It is also possible to study the solutions of Fermat type equations over
number fields asymptotically. In this talk\, I will mention some recent re
sults about these notions for the classical Fermat equation as well as som
e other Diophantine equations.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College London)
DTSTART;VALUE=DATE-TIME:20220407T150000Z
DTEND;VALUE=DATE-TIME:20220407T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/120
DESCRIPTION:Title: On the cohomology of Shimura varieties with torsion coefficients\nby Ana Caraiani (Imperial College London) as part of Number Theory Web
Seminar\n\n\nAbstract\nShimura varieties are certain highly symmetric alge
braic varieties that generalise modular curves and that play an important
role in the Langlands program. In this talk\, I will survey recent vanishi
ng conjectures and results about the cohomology of Shimura varieties with
torsion coefficients\, under both local and global representation-theoreti
c conditions. I will illustrate the geometric ingredients needed to establ
ish these results using the toy model of the modular curve. I will also me
ntion several applications\, including to (potential) modularity over CM f
ields.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Larry Guth (MIT)
DTSTART;VALUE=DATE-TIME:20220127T160000Z
DTEND;VALUE=DATE-TIME:20220127T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/121
DESCRIPTION:Title: Reflections on the proof(s) of the Vinogradov mean value conjecture
\nby Larry Guth (MIT) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nThe Vinogradov mean value conjecture concerns the number of solutions
of a system of diophantine equations. This number of solutions can also b
e written as a certain moment of a trigonometric polynomial. The conjectu
re was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley\, and re
cently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The details
of each proof involve some intricate estimates. The goal of the talk is
to try to reflect on the proof(s) in a big picture way. A key ingredient
in all the proofs is to combine estimates at many different scales\, usual
ly by doing induction on scales. Why does this multi-scale induction help
? What can multi-scale induction tell us and what are its limitations?\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Humphries (University of Virginia)
DTSTART;VALUE=DATE-TIME:20220203T160000Z
DTEND;VALUE=DATE-TIME:20220203T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/122
DESCRIPTION:Title: $L^p$-norm bounds for automorphic forms\nby Peter Humphries (Un
iversity of Virginia) as part of Number Theory Web Seminar\n\n\nAbstract\n
A major area of study in analysis involves the distribution of mass of Lap
lacian eigenfunctions on a Riemannian manifold. A key result towards this
is explicit $L^p$-norm bounds for Laplacian eigenfunctions in terms of the
ir Laplacian eigenvalue\, due to Sogge in 1988. Sogge's bounds are sharp o
n the sphere\, but need not be sharp on other manifolds. I will discuss so
me aspects of this problem for the modular surface\; in this setting\, the
Laplacian eigenfunctions are automorphic forms\, and certain $L^p$-norms
can be shown to be closely related to certain mixed moments of $L$-functio
ns. This is joint with with Rizwanur Khan.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ram Murty (Queen's University)
DTSTART;VALUE=DATE-TIME:20220414T150000Z
DTEND;VALUE=DATE-TIME:20220414T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/124
DESCRIPTION:Title: Probability Theory and the Riemann Hypothesis\nby Ram Murty (Qu
een's University) as part of Number Theory Web Seminar\n\n\nAbstract\nTher
e is a probability distribution attached to the Riemann zeta function whic
h allows one to formulate the Riemann hypothesis in terms of the cumulants
of this distribution and is due to Biane\, Pitman and Yor. The cumulants
can be related to generalized Euler-Stieltjes constants and to Li's criter
ion for the Riemann hypothesis. We will discuss these results and present
some new results related to this theme.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jozsef Solymosi (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20220120T160000Z
DTEND;VALUE=DATE-TIME:20220120T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/125
DESCRIPTION:Title: Rank of matrices with entries from a multiplicative group\nby J
ozsef Solymosi (University of British Columbia) as part of Number Theory W
eb Seminar\n\n\nAbstract\nWe establish lower bounds on the rank of matrice
s in which all but the diagonal entries lie in a multiplicative group of s
mall rank. Applying these bounds we show that the distance sets of finite
pointsets in $\\R^d$ generate high rank multiplicative groups and that mul
tiplicative groups of small rank cannot contain large sumsets. (Joint work
with Noga Alon)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Kleinbock (Brandeis University)
DTSTART;VALUE=DATE-TIME:20220310T160000Z
DTEND;VALUE=DATE-TIME:20220310T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/126
DESCRIPTION:Title: Shrinking targets on homogeneous spaces and improving Dirichlet's T
heorem\nby Dmitry Kleinbock (Brandeis University) as part of Number Th
eory Web Seminar\n\n\nAbstract\nLet $\\psi$ be a decreasing function defin
ed on all large positive real numbers. We say that a real $m \\times n$ ma
trix $Y$ is "$\\psi$-Dirichlet" if for every sufficiently large real numbe
r $T$ there exist non-trivial integer vectors $(p\,q)$ satisfying $\\|Yq-p
\\|^m < \\psi(T)$ and $\\|q\\|^n < T$ (where $\\|\\cdot\\|$ denotes the su
premum norm on vectors). This generalizes the property of $Y$ being "Diric
hlet improvable" which has been studied by several people\, starting with
Davenport and Schmidt in 1969. I will present results giving sufficient co
nditions on $\\psi$ to ensure that the set of $\\psi$-Dirichlet matrices h
as zero (resp.\, full) measure. If time allows I will mention a geometric
generalization of the set-up\, where the supremum norm is replaced by an a
rbitrary norm. Joint work with Anurag Rao\, Andreas Strombergsson\, Nick W
adleigh and Shuchweng Yu.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Levin (Michigan State University)
DTSTART;VALUE=DATE-TIME:20220317T160000Z
DTEND;VALUE=DATE-TIME:20220317T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/127
DESCRIPTION:Title: Diophantine Approximation for Closed Subschemes\nby Aaron Levin
(Michigan State University) as part of Number Theory Web Seminar\n\n\nAbs
tract\nThe classical Weil height machine associates heights to divisors on
a projective variety. I will give a brief\, but gentle\, introduction to
how this machinery extends to objects (closed subschemes) in higher codime
nsion\, due to Silverman\, and discuss various ways to interpret the heigh
ts. We will then discuss several recent results in which these ideas play
a prominent and central role.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Granville (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20220428T150000Z
DTEND;VALUE=DATE-TIME:20220428T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/128
DESCRIPTION:Title: Linear Divisibility sequences\nby Andrew Granville (Université
de Montréal) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1878
\, in the first volume of the first mathematics journal published in the U
S\, Edouard Lucas wrote 88 pages (in French) on linear recurrence sequence
s\, placing Fibonacci numbers and other linear recurrence sequences into a
broader context. He examined their behaviour locally as well as globally\
, and asked several questions that influenced much research in the century
and a half to come.\n\nIn a sequence of papers in the 1930s\, Marshall Ha
ll further developed several of Lucas' themes\, including studying and try
ing to classify third order linear divisibility sequences\; that is\, line
ar recurrences like the Fibonacci numbers which have the additional proper
ty that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many spec
ial cases\, Hall was unable to even conjecture what a general theorem sho
uld look like\, and despite developments over the years by various authors
\, such as Lehmer\, Morgan Ward\, van der Poorten\, Bezivin\, Petho\, Rich
ard Guy\, Hugh Williams\,... with higher order linear divisibility sequenc
es\, even the formulation of the classification has remained mysterious.\n
\nIn this talk we present our ongoing efforts to classify all linear divis
ibility sequences\, the key new input coming from a wonderful application
of the Schmidt/Schlickewei subspace theorem from the theory of diophantine
approximation\, due to Corvaja and Zannier.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harry Schmidt (University of Basel)
DTSTART;VALUE=DATE-TIME:20220217T160000Z
DTEND;VALUE=DATE-TIME:20220217T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/129
DESCRIPTION:Title: Counting rational points and lower bounds for Galois orbits for spe
cial points on Shimura varieties\nby Harry Schmidt (University of Base
l) as part of Number Theory Web Seminar\n\n\nAbstract\nIn this talk I will
give an overview of the history of the André-Oort conjecture and its res
olution last year after the final steps were made in work of Pila\, Shanka
r\, Tsimerman\, Esnault and Groechenig as well as Binyamini\, Yafaev and m
yself. I will focus on the key insights and ideas related to model theory
and transcendence theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART;VALUE=DATE-TIME:20220505T150000Z
DTEND;VALUE=DATE-TIME:20220505T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/132
DESCRIPTION:Title: On integers which are(n't) the sum of two rational cubes\nby Le
vent Alpöge (Harvard University) as part of Number Theory Web Seminar\n\n
\nAbstract\nIt's easy that $0\\%$ of integers are the sum of two integral
cubes (allowing opposite signs!).\nI will explain joint work with Bhargava
and Shnidman in which we show:\n\n1. At least a sixth of integers are not
the sum of two rational cubes\,\n\nand\n\n2. At least a sixth of odd inte
gers are the sum of two rational cubes!\n(--- with 2. relying on new $2$-c
onverse results of Burungale-Skinner.)\n\nThe basic principle is that "the
re aren't even enough $2$-Selmer elements to go around" to contradict e.g.
1.\, and we show this by using the circle method "inside" the usual geome
try of numbers argument applied to a particular coregular representation.
Even then the resulting constant isn't small enough to conclude 1.\, so we
use the clean form of root numbers in the family $x^3 + y^3 = n$ and the
$p$-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winnie Li (Pennsylvania State University)
DTSTART;VALUE=DATE-TIME:20220324T160000Z
DTEND;VALUE=DATE-TIME:20220324T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/133
DESCRIPTION:Title: Group based zeta functions\nby Winnie Li (Pennsylvania State Un
iversity) as part of Number Theory Web Seminar\n\n\nAbstract\nThe theme of
this survey talk is zeta functions which count closed geodesics on object
s arising from real and $p$-adic groups. Our focus is on $\\PGL(n)$. For $
\\PGL(2)$\, these are the Selberg zeta function for compact quotients of t
he upper half-plane and the Ihara zeta function for finite regular graphs.
We shall explain the identities satisfied by these zeta functions\, which
show interconnections between combinatorics\, group theory and number the
ory. Comparisons will be made for zeta identities from different backgroun
d. Like the Riemann zeta function\, the analytic behavior of a group base
d zeta function governs the distribution of the prime geodesics in its def
inition.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teräväinen (University of Turku)
DTSTART;VALUE=DATE-TIME:20220421T150000Z
DTEND;VALUE=DATE-TIME:20220421T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/134
DESCRIPTION:Title: Short exponential sums of the primes\nby Joni Teräväinen (Uni
versity of Turku) as part of Number Theory Web Seminar\n\n\nAbstract\nI wi
ll discuss the short interval behaviour of the von Mangoldt and Möbius fu
nctions twisted by exponentials. I will in particular mention new results
on sums of these functions twisted by polynomial exponential phases\, or e
ven more general nilsequence phases. I will also discuss connections to Ch
owla's conjecture. This is based on joint works with Kaisa Matomäki\, Mak
sym Radziwiłł\, Xuancheng Shao\, Terence Tao and Tamar Ziegler.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Chen (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20220331T150000Z
DTEND;VALUE=DATE-TIME:20220331T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/135
DESCRIPTION:Title: Markoff triples and connectivity of Hurwitz spaces\nby William
Chen (Institute for Advanced Study) as part of Number Theory Web Seminar\n
\n\nAbstract\nIn this talk we will show that the integral points of the Ma
rkoff equation $x^2 + y^2 + z^2 - xyz = 0$ surject onto its $F_p$-points f
or all but finitely many primes $p$. This essentially resolves a conjectur
e of Bourgain\, Gamburd\, and Sarnak\, and a question of Frobenius from 19
13. The proof relates the question to the classical problem of classifying
the connected components of the Hurwitz moduli spaces $H(g\,n)$ classifyi
ng finite covers of genus $g$ curves with $n$ branch points. Over a centur
y ago\, Clebsch and Hurwitz established connectivity for the subspace clas
sifying simply branched covers of the projective line\, which led to the f
irst proof of the irreducibility of the moduli space of curves of a given
genus. More recently\, the work of Dunfield-Thurston and Conway-Parker est
ablish connectivity in certain situations where the monodromy group is fix
ed and either $g$ or $n$ are allowed to be large\, which has been applied
to study Cohen-Lenstra heuristics over function fields. In the case where
$(g\,n)$ are fixed and the monodromy group is allowed to vary\, far less i
s known. In our case we study $\\SL(2\,p)$-covers of elliptic curves\, onl
y branched over the origin\, and establish connectivity\, for all sufficie
ntly large p\, of the subspace classifying those covers with ramification
indices $2p$. The proof builds upon asymptotic results of Bourgain\, Gambu
rd\, and Sarnak\, the key new ingredient being a divisibility result on th
e degree of a certain forgetful map between moduli spaces\, which provides
enough rigidity to bootstrap their asymptotics to a result for all suffic
iently large $p$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elon Lindenstrauss (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20220602T150000Z
DTEND;VALUE=DATE-TIME:20220602T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/136
DESCRIPTION:Title: Effective equidistribution of some unipotent flows with polynomial
rates\nby Elon Lindenstrauss (Hebrew University of Jerusalem) as part
of Number Theory Web Seminar\n\n\nAbstract\nJoint work with Amir Mohammadi
and Zhiren Wang\n\nA landmark result of Ratner gives that if $G$ is a rea
l linear algebraic group\, $\\Gamma$ a lattice in $G$ and if $u_t$ is a on
e-parameter unipotent subgroup of $G$\, then for any $x \\in G/\\Gamma$ th
e orbit $u_t.x$ is equidistributed in a periodic orbit of some subgroup $L
< G$\, and moreover that the orbit of $x$ under $u_t$ is contained in thi
s periodic $L$ orbit.\n\nA key motivation behind Ratner's equidistribution
theorem for one-parameter unipotent flows has been to establish Raghunath
an's conjecture regarding the possible orbit closures of groups generated
by one-parameter unipotent groups\; using the equidistribution theorem Rat
ner proved that if $G$ and $\\Gamma$ are as above\, and if $H < G$ is gene
rated by one parameter unipotent groups then for any $x \\in G/\\Gamma$ on
e has that $\\overline{H.x}=L.x$ where $H < L < G$ and $L.x$ is periodic.
Important special cases of Raghunathan's conjecture were proven earlier by
Margulis and by Dani and Margulis by a different\, more direct\, approach
.\n\nThese results have had many beautiful and unexpected applications in
number theory\, geometry and other areas. A key challenge has been to quan
tify and effectify these results. Beyond the case of actions of horospheri
c groups where there are several fully quantitative and effective results
available\, results in this direction have been few and far between. In pa
rticular\, if $G$ is semisimple and $U$ is not horospheric no quantitative
form of Ratner's equidistribution was known with any error rate\, though
there has been some progress on understanding quantitatively density prope
rties of such flows with iterative logarithm error rates.\n\nIn my talk I
will present a new fully quantitative and effective equidistribution resul
t for orbits of one-parameter unipotent groups in arithmetic quotients of
$\\SL_2(\\C)$ and $\\SL_2(\\R)\\times\\SL(2\,\\R)$. I will also try to exp
lain a bit the connection to number theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Princeton University)
DTSTART;VALUE=DATE-TIME:20220526T150000Z
DTEND;VALUE=DATE-TIME:20220526T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/137
DESCRIPTION:Title: Applications of arithmetic holonomicity theorems\nby Yunqing Ta
ng (Princeton University) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nIn this talk\, we will discuss the proof of the unbounded denominators
conjecture on Fourier coefficients of $\\SL_2(\\Z)$-modular forms\, and t
he proof of irrationality of $2$-adic zeta value at $5$. Both proofs use a
n arithmetic holonomicity theorem\, which can be viewed as a refinement of
André’s algebraicity criterion. If time permits\, we will give a proof
of the arithmetic holonomicity theorem via the slope method a la Bost.\nT
his is joint work with Frank Calegari and Vesselin Dimitrov.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20220210T160000Z
DTEND;VALUE=DATE-TIME:20220210T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/138
DESCRIPTION:Title: Beyond uniform distribution\nby Zeev Rudnick (Tel Aviv Universi
ty) as part of Number Theory Web Seminar\n\n\nAbstract\nThe study of unifo
rm distribution of sequences is more than a century old\, with pioneering
work by Hardy and Littlewood\, Weyl\, van der Corput and others. More rece
ntly\, the focus of research has shifted to much finer quantities\, such a
s the distribution of nearest neighbor gaps and the pair correlation funct
ion. Examples of interesting sequences for which these quantities have bee
n studied include the zeros of the Riemann zeta function\, energy levels o
f quantum systems\, and more. In this expository talk\, I will discuss wha
t is known about these examples and discuss the many outstanding problems
that this theory has to offer.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Looper (Brown University)
DTSTART;VALUE=DATE-TIME:20220609T150000Z
DTEND;VALUE=DATE-TIME:20220609T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/139
DESCRIPTION:Title: The Uniform Boundedness Principle for polynomials over number field
s\nby Nicole Looper (Brown University) as part of Number Theory Web Se
minar\n\n\nAbstract\nThis talk is about uniform bounds on the number of $K
$-rational preperiodic points across families of endomorphisms of projecti
ve space defined over various fields $K$. We will focus on the case where
$K$ is a number field\, and the morphisms are polynomial maps on $\\mathbb
{P}^1$. Along the way\, I will highlight the more challenging aspects behi
nd the known approaches\, and discuss the obstacles to be addressed in fut
ure research.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Shpilka (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20220623T150000Z
DTEND;VALUE=DATE-TIME:20220623T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/140
DESCRIPTION:Title: Points\, lines and polynomial identities\nby Amir Shpilka (Tel
Aviv University) as part of Number Theory Web Seminar\n\n\nAbstract\nThe S
ylvester-Gallai (SG) theorem in discrete geometry asserts that if a finite
set of points P has the property that every line through any two of its p
oints intersects the set at a third point\, then P must lie on a line. Sur
prisingly\, this theorem\, and some variants of it\, appear in the analysi
s of locally correctable codes and\, more noticeably\, in algebraic progra
m testing (polynomial identity testing). For these questions one often has
to study extensions of the original SG problem: the case where there are
several sets\, or with a robust version of the condition (many "special" l
ines through each point) or with a higher degree analog of the problem\, e
tc.\n\nIn this talk I will present the SG theorem and some of its variants
\, show its relation to the above mentioned computational problems and dis
cuss recent developments regarding higher degree analogs and their applica
tions.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART;VALUE=DATE-TIME:20220616T150000Z
DTEND;VALUE=DATE-TIME:20220616T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/141
DESCRIPTION:Title: Counting elliptic curves with level structure\nby John Voight (
Dartmouth College) as part of Number Theory Web Seminar\n\n\nAbstract\nFol
lowing work of Harron and Snowden\, we provide an asymptotic answer to que
stions like: how many elliptic curves of bounded height have a cyclic isog
eny of degree $N$? We'll begin\nwith a survey the recent spate of work on
this topic\, and then we will report on joint work with Carl Pomerance and
Maggie Pizzo\, with John Cullinan and Meagan Kenney\, and finally\nwith G
rant Molnar.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW Sydney)
DTSTART;VALUE=DATE-TIME:20220224T160000Z
DTEND;VALUE=DATE-TIME:20220224T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/142
DESCRIPTION:Title: Sums of Kloosterman and Salie Sums and Moments of $L$-functions\nby Igor Shparlinski (UNSW Sydney) as part of Number Theory Web Seminar\
n\n\nAbstract\nWe present some old and more recent results which suggest t
hat Kloosterman and Salie sums exhibit a pseudorandom behaviour similar to
the behaviour which is traditionally attributed to the Mobius function. I
n particular\, we formulate some analogues of the Chowla Conjecture for Kl
oosterman and Salie sums. We then describe several results about the non-c
orrelation of Kloosterman and Salie sums between themselves and also with
some classical number-theoretic functions such as the Mobius function\, th
e divisor function and the sums of binary digits. Various arithmetic appli
cations of these results\, including to asymptotic formulas for moments of
various $L$-functions\, will be outlined as well.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Gamburd (CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20220630T150000Z
DTEND;VALUE=DATE-TIME:20220630T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/143
DESCRIPTION:Title: Arithmetic and dynamics on varieties of Markoff type\nby Alexan
der Gamburd (CUNY Graduate Center) as part of Number Theory Web Seminar\n\
n\nAbstract\nThe Markoff equation $x^2+y^2+z^2=3xyz$\, which arose in his
spectacular thesis (1879)\, is ubiquitous in a tremendous variety of conte
xts. After reviewing some of these\, we will discuss recent progress towar
ds establishing forms of strong approximation on varieties of Markoff type
\, as well as ensuing implications\, diophantine and dynamical.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Vaaler (University of Texas at Austin)
DTSTART;VALUE=DATE-TIME:20220519T150000Z
DTEND;VALUE=DATE-TIME:20220519T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/144
DESCRIPTION:Title: Schinzel's determinant inequality and a conjecture of F. Rodriguez
Villegas\nby Jeffrey Vaaler (University of Texas at Austin) as part of
Number Theory Web Seminar\n\n\nAbstract\nThe Abstract is available at\n\n
https://www.ntwebseminar.org/home\n\nor directly at\n\nhttps://drive.googl
e.com/file/d/1VDQLDlcC3IDEMduR6H-X9Rf0jRxSZ_J-/view\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Charles Vaughan (Pennsylvania State University)
DTSTART;VALUE=DATE-TIME:20220512T150000Z
DTEND;VALUE=DATE-TIME:20220512T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/145
DESCRIPTION:Title: Generalizations of the Montgomery-Hooley asymptotic formula\; A sur
vey.\nby Robert Charles Vaughan (Pennsylvania State University) as par
t of Number Theory Web Seminar\n\n\nAbstract\nFollowing a statement withou
t proof in a special case by Barban [1966]\, and less precise bounds by Da
venport and Halberstam [1966] and Gallagher [1967]\, Montgomery [1970] obt
ained the asymptotic formula\n\\[\n\\sum_{q\\le Q} \\sum_{\\stackrel{a=1}{
(a\,q)=1}}^q \\left|\n\\psi(x\;q\,a) - \\frac{x}{\\phi(q)}\n\\right|^2 \\s
im Qx\\log x\n\\]\nvalid when $x(\\log x)^{-A}\\le Q\\le x$ and $A$ is fix
ed. This was refined and the proof substantially simplified by Hooley [19
75] in the first of a celebrated series of 19 papers with the generic titl
e ``On the Barban-Davenport-Halberstam theorem" which have widened the sco
pe of the methods. There have been also a number of papers by other autho
rs with further generalizations and I will give a survey of this together
with an overview of some of the recent developments.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (University of California Irvine)
DTSTART;VALUE=DATE-TIME:20220922T150000Z
DTEND;VALUE=DATE-TIME:20220922T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/146
DESCRIPTION:Title: Negative moments of the Riemann zeta function\nby Alexandra Flo
rea (University of California Irvine) as part of Number Theory Web Seminar
\n\n\nAbstract\nI will talk about work towards a conjecture of Gonek regar
ding negative shifted moments of the Riemann zeta function. I will explain
how to obtain asymptotic formulas when the shift in the Riemann zeta func
tion is big enough\, and how one can obtain non-trivial upper bounds for s
maller shifts. Joint work with H. Bui.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ping Xi (Xi'an Jiaotong University)
DTSTART;VALUE=DATE-TIME:20220908T150000Z
DTEND;VALUE=DATE-TIME:20220908T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/147
DESCRIPTION:Title: Analytic approaches towards Katz’s problems on Kloosterman sums\nby Ping Xi (Xi'an Jiaotong University) as part of Number Theory Web Se
minar\n\n\nAbstract\nMotivated by deep observations on elliptic curves/mod
ular forms\, Nicholas Katz proposed three problems on sign changes\, equid
istributions and modular structures of Kloosterman sums in 1980. In this t
alk\, we will discuss some recent progresses towards these three problems
made by analytic number theory (e.g.\, sieve methods and automorphic forms
) combining certain tools from $\\ell$-adic cohomology.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yann Bugeaud (University of Strasbourg)
DTSTART;VALUE=DATE-TIME:20220901T150000Z
DTEND;VALUE=DATE-TIME:20220901T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/148
DESCRIPTION:Title: $B'$\nby Yann Bugeaud (University of Strasbourg) as part of Num
ber Theory Web Seminar\n\n\nAbstract\nLet $n \\ge 1$ be an integer and $\\
alpha_1\, \\ldots\, \\alpha_n$ be non-zero algebraic numbers. \nLet $b_1\,
\\ldots \, b_n$ be integers with $b_n \\not= 0$\, and set $B = \\max\\{3\
, |b_1|\, \\ldots \, |b_n|\\}$. \nFor $j =1\, \\ldots\, n$\, set $h^* (\\a
lpha_j) = \\max\\{h(\\alpha_j)\, 2\\}$\, where $h$ \ndenotes the (logarith
mic) Weil height. \nAssume that the quantity $\\Lambda = b_1 \\log \\alpha
_1 + \\cdots + b_n \\log \\alpha_n$ is nonzero. \nA typical lower bound of
$\\log |\\Lambda|$ given by Baker's theory of linear forms in logarithms
takes the shape \n$$\n- c(n\, D) \\\, h^* (\\alpha_1) \\ldots h^*(\\alph
a_n) \\log B\, \n$$\nwhere $c(n\,D)$ is positive\, effectively computable
and depends only on $n$ and on the degree $D$ of the field generated \nby
$\\alpha_1\, \\ldots \, \\alpha_n$. \nHowever\, in certain special cases a
nd in particular when $|b_n| = 1$\, this bound can be improved to\n$$\n- c
(n\, D) \\\, h^* (\\alpha_1) \\ldots h^*(\\alpha_n) \\log \\frac{B}{h^*
(\\alpha_n)}.\n$$\nThe term $B' := B / h^*(\\alpha_n)$ in place of $B$ \no
riginates in works of Feldman and of Baker. It is a key tool for improving
\, in an effective way\, the upper bound for the irrationality exponent\no
f a real algebraic number of degree at least $3$ given by \nLiouville's th
eorem.\nWe survey various applications of this $B'$ to exponents of approx
imation evaluated at algebraic numbers\, \nto the $S$-part of integer sequ
ences\, and to Diophantine equations.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Nelson (Aarhus University)
DTSTART;VALUE=DATE-TIME:20220929T150000Z
DTEND;VALUE=DATE-TIME:20220929T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/149
DESCRIPTION:Title: The sup norm problem in the level aspect\nby Paul Nelson (Aarhu
s University) as part of Number Theory Web Seminar\n\n\nAbstract\nThe sup
norm problem concerns the size of $L^2$-normalized eigenfunctions of manif
olds. In many situations\, one expects to be able to improve upon the gen
eral bound following from local considerations. The pioneering result in
that direction is due to Iwaniec and Sarnak\, who in 1995 established an i
mprovement upon the local bound for Hecke-Maass forms of large eigenvalue
on the modular surface. Their method has since been extended and applied
by many authors\, notably to the "level aspect" variant of the problem\, w
here one varies the underlying manifold rather than the eigenvalue. Recen
tly\, Raphael Steiner introduced a new method for attacking the sup norm p
roblem. I will describe joint work with Raphael Steiner and Ilya Khayutin
in which we apply that method to improve upon the best known bounds in th
e level aspect.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey C. Lagarias (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221006T150000Z
DTEND;VALUE=DATE-TIME:20221006T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/150
DESCRIPTION:Title: The Alternative Hypothesis and Point Processes\nby Jeffrey C. L
agarias (University of Michigan) as part of Number Theory Web Seminar\n\n\
nAbstract\nThe Alternative Hypothesis concerns a hypothetical and unlikely
picture of how zeros of the Riemann zeta function are spaced. It asks tha
t nearly all normalized zero spacings be near half-integers. This possi
ble zero distribution is incompatible with the GUE distribution of zero sp
acings. Ruling it out arose as an obstacle to the long-standing problem o
f proving there are no exceptional zeros of Dirichlet $L$-functions. The
talk describes joint work with Brad Rodgers\, that constructs a point proc
ess realizing Alternative Hypothesis type statistics\, which is consiste
nt with the known results on correlation functions for spacings of zeta z
eros. (A similar result was independently obtained by Tao with slightly d
ifferent methods.) The talk reviews point process models and presents fur
ther results on the general problem of to what extent two point processes\
, a continuous one on the real line\, the other a discrete one on a lattic
e $a\\Z$\, can mimic each other in the sense of having perfect agreement
of all their correlation functions when convolved with bandlimited test fu
nctions of a given bandwidth $B$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Evra (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20221103T160000Z
DTEND;VALUE=DATE-TIME:20221103T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/151
DESCRIPTION:Title: Optimal strong approximation and the Sarnak-Xue density hypothesis<
/a>\nby Shai Evra (Hebrew University of Jerusalem) as part of Number Theor
y Web Seminar\n\n\nAbstract\nIt is a classical result that the modulo map
from $\\SL_2(\\Z)$ to $\\SL_2(\\Z/q\\Z)$\, is surjective for any integer $
q$. The generalization of this phenomenon to other arithmetic groups goes
under the name of strong approximation\, and it is well understood. The fo
llowing natural question was recently raised in a letter of Sarnak: What i
s the minimal exponent $e$\, such that for any large $q$\, almost any elem
ent of $\\SL_2(\\Z/q\\Z)$ has a lift in $\\SL_2(\\Z)$ with coefficients of
size at most $q^e$? A simple pigeonhole principle shows that $e > 3/2$. I
n his letter Sarnak proved that this is in fact tight\, namely $e = 3/2$\,
and call this optimal strong approximation for $\\SL_2(\\Z)$. The proof r
elies on a density theorem of the Ramanujan conjecture for $\\SL_2(\\Z)$.\
n\nIn this talk we will give a brief overview of the strong approximation\
, a quantitative strengthening of it called super strong approximation\, a
nd the above mentioned optimal strong approximation phenomena\, for arithm
etic groups. We highlight the special case of $p$-arithmetic subgroups of
classical definite matrix groups and the connection between the optimal st
rong approximation and optimal almost diameter for Ramanujan complexes. Fi
nally\, we will present the Sarnak-Xue density hypothesis and describe rec
ent ongoing works on it relying on deep results coming from the Langlands
program.\n\nThis talk is based on ongoing joint works with B. Feigon\, M.
Gerbelli-Gauthier\, H. Gustafssun\, K. Maurischat and O. Parzanchevski.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evelina Viada (University of Göttingen)
DTSTART;VALUE=DATE-TIME:20221027T150000Z
DTEND;VALUE=DATE-TIME:20221027T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/152
DESCRIPTION:Title: Rational points on curves in a product of elliptic curves\nby E
velina Viada (University of Göttingen) as part of Number Theory Web Semin
ar\n\n\nAbstract\nThe Mordell-Conjecture (Faltings Theorem) states that an
algebraic curve of genus at least $2$ has only finitely many rational poi
nts. The Torsion Anomalous Conjecture (TAC) generalises Faltings Theorem.
In some cases the proofs of the TAC are effective\, implying effective cas
es of the Mordell-Conjecture. I would like to explain an effective method
to determine the $K$-rational points on certain families of curves and to
present some new specific examples. I will give an overview of the methods
used in the context of the TAC presenting some general theorems and appl
ications.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Neftin (Technion-Israel Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220915T150000Z
DTEND;VALUE=DATE-TIME:20220915T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/153
DESCRIPTION:Title: Reducible fibers of polynomial maps\nby Danny Neftin (Technion-
Israel Institute of Technology) as part of Number Theory Web Seminar\n\n\n
Abstract\nFor a polynomial $f\\in \\mathbb Q[x]$\, the fiber $f^{-1}(a)$ i
s irreducible over $\\mathbb Q$ for all values $a\\in \\mathbb Q$ outside
a ``thin" set of exceptions $R_f$ whose explicit description is unknown in
general. The problem of describing $R_f$ is closely related to reducibili
ty and arboreal representations in arithmetic dynamics\, as well as to Kro
necker and arithmetic equivalence for polynomial maps\, that is\, polynomi
al versions of the question: "can you hear the shape of the drum?". We sha
ll discuss recent progress on the above problem and topics.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Gauthier (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20221013T150000Z
DTEND;VALUE=DATE-TIME:20221013T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/154
DESCRIPTION:Title: A complex analytic approach to sparsity\, rigidity and uniformity i
n arithmetic dynamics\nby Thomas Gauthier (Université Paris-Saclay) a
s part of Number Theory Web Seminar\n\n\nAbstract\nThis talk is concerned
with connections between arithmetic dynamics and complex dynamics. The fir
st aim of the talk is to discuss several open problems from arithmetic dyn
amics and to explain how these problems are related to complex dynamical t
ool: bifurcation measures.\nIf time allows\, I will give a strategy to tac
kle several of those problems at the same time. This is based on a joint w
ork in progress with Gabriel Vigny and Johan Taflin.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20221020T150000Z
DTEND;VALUE=DATE-TIME:20221020T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/155
DESCRIPTION:Title: Symmetric power functoriality for $\\GL(2)$\nby Jack Thorne (Un
iversity of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstract\
nLanglands’s functoriality conjectures predict the existence of “lifti
ngs” of automorphic representations along morphisms of $L$-groups. A bas
ic case of interest comes from the irreducible algebraic representations o
f $\\GL(2)$ – the associated symmetric power $L$-functions are then the
ones identified by Serre in the 1960’s in relation to the Sato—Tate co
njecture.\n\nI will describe the background to these ideas and then discus
s the proof\, joint with James Newton\, of the existence of these symmetri
c power liftings for Hilbert modular forms. One arithmetic consequence is
that if $E$ is a (non-CM) elliptic curve over a real quadratic field\, the
n all of its symmetric power $L$-functions admit analytic continuation to
the whole complex plane.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuel Carneiro (ICTP)
DTSTART;VALUE=DATE-TIME:20221110T160000Z
DTEND;VALUE=DATE-TIME:20221110T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/156
DESCRIPTION:Title: Hilbert spaces and low-lying zeros of $L$-functions\nby Emanuel
Carneiro (ICTP) as part of Number Theory Web Seminar\n\n\nAbstract\nIn th
is talk I would like to present some ideas behind a general Hilbert space
framework for solving certain optimization problems that arise when studyi
ng the distribution of the low-lying zeros of families of $L$-functions. F
or instance\, in connection to previous work of Iwaniec\, Luo\, and Sarnak
(2000)\, we will discuss how to use information from one-level density th
eorems to estimate the proportion of non-vanishing of $L$-functions in a f
amily at a low-lying height on the critical line. We will also discuss the
problem of estimating the height of the first low-lying zero in a family\
, considered by Hughes and Rudnick (2003) and Bernard (2015). This is base
d on joint work with M. Milinovich and A. Chirre.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART;VALUE=DATE-TIME:20221117T160000Z
DTEND;VALUE=DATE-TIME:20221117T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/157
DESCRIPTION:Title: Waring’s Problem\nby Trevor Wooley (Purdue University) as par
t of Number Theory Web Seminar\n\n\nAbstract\nIn 1770\, E. Waring made an
assertion these days interpreted as conjecturing that when $k$ is a natura
l number\, all positive integers may be written as the sum of a number $g(
k)$ of positive integral $k$-th powers\, with $g(k)$ finite. Since the wor
k of Hardy and Littlewood a century ago\, attention has largely shifted to
the problem of bounding $G(k)$\, the least number $s$ having the property
that all sufficiently large integers can be written as the sum of $s$ pos
itive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange)\, $G(3)
\\le 7$ (Linnik)\, $G(4)=16$ (Davenport)\, and $G(5)\\le 17$\, $G(6)\\le 2
4$\, ...\, $G(20)\\le 142$ (Vaughan and Wooley). For large $k$ one has $G(
k)\\le k(\\log k+\\log \\log k+2+o(1))$ (Wooley). We report on very recent
progress joint with Joerg Bruedern. One or two new world records will be
on display.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (University of Oxford)
DTSTART;VALUE=DATE-TIME:20221124T160000Z
DTEND;VALUE=DATE-TIME:20221124T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/158
DESCRIPTION:Title: A proof of the Erdős primitive set conjecture\nby Jared Duker
Lichtman (University of Oxford) as part of Number Theory Web Seminar\n\n\n
Abstract\nA set of integers greater than 1 is primitive if no member in th
e set divides another. Erdős proved in 1935 that the sum of 1/(a log a)\,
ranging over a in A\, is uniformly bounded over all choices of primitive
sets A. In 1986 he asked if this bound is attained for the set of prime nu
mbers. In this talk we describe recent work which answers Erdős’ conjec
ture in the affirmative. We will also discuss applications to old question
s of Erdős\, Sárközy\, and Szemerédi from the 1960s.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert J. Lemke Oliver (Tufts University)
DTSTART;VALUE=DATE-TIME:20221201T160000Z
DTEND;VALUE=DATE-TIME:20221201T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/159
DESCRIPTION:Title: Uniform exponent bounds on the number of primitive extensions of nu
mber fields\nby Robert J. Lemke Oliver (Tufts University) as part of N
umber Theory Web Seminar\n\n\nAbstract\nA folklore conjecture asserts the
existence of a constant $c_n > 0$ such that $N_n(X) \\sim c_n X$ as $X\\to
\\infty$\, where $N_n(X)$ is the number of degree $n$ extensions $K/\\mat
hbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n
\\leq 5$\, but even the weaker conjecture that there exists an absolute co
nstant $C\\geq 1$ such that $N_n(X) \\ll_n X^C$ remains unknown and appare
ntly out of reach.\n\nHere\, we make progress on this weaker conjecture (w
hich we term the ``uniform exponent conjecture'') in two ways. First\, we
reduce the general problem to that of studying relative extensions of num
ber fields whose Galois group is an almost simple group in its smallest de
gree permutation representation. Second\, for almost all such groups\, we
prove the strongest known upper bound on the number of such extensions.
These bounds have the effect of resolving the uniform exponent conjecture
for solvable groups\, sporadic groups\, exceptional groups\, and classical
groups of bounded rank. This is forthcoming work that grew out of conver
sations with M. Bhargava.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART;VALUE=DATE-TIME:20221208T160000Z
DTEND;VALUE=DATE-TIME:20221208T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/160
DESCRIPTION:Title: Lattès maps\, bifurcations\, and arithmetic\nby Laura DeMarco
(Harvard University) as part of Number Theory Web Seminar\n\n\nAbstract\nI
n the field of holomorphic dynamics\, we learn that the Lattès maps -- th
e rational functions on $\\mathbb{P}^1$ that are quotients of maps on elli
ptic curves -- are rather boring. We can understand their dynamics comple
tely. But viewed arithmetically\, there are still unanswered questions.
I'll begin the talk with some history of these maps. Then I'll describe o
ne of the recent questions and how it has led to interesting complex-dynam
ical questions about other families of maps on $\\mathbb{P}^1$ and\, in tu
rn\, new perspectives on the arithmetic side. The new material is a joint
project with Myrto Mavraki.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore Pisa)
DTSTART;VALUE=DATE-TIME:20221222T160000Z
DTEND;VALUE=DATE-TIME:20221222T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/161
DESCRIPTION:Title: Bounded generation in linear groups and exponential parametrization
s\nby Umberto Zannier (Scuola Normale Superiore Pisa) as part of Numbe
r Theory Web Seminar\n\n\nAbstract\nIn fairly recent joint work with Corva
ja\, Rapinchuk\, Ren\, we applied results from Diophantine S-unit theory t
o problems of “bounded generation” in linear groups: this property is
a strong form of finite generation and is useful for several issues in the
setting. Focusing on “anisotropic groups” (i.e. containing only semi-
simple elements)\, we could give a simple essentially complete description
of those with the property. More recently\, in further joint work also wi
th Demeio\, we proved the natural expectation that sets boundedly generate
d by semi-simple elements (in linear groups over number fields) are “sp
arse”. Actually\, this holds for all sets obtained by exponential parame
trizations. As a special consequence\, this gives back the previous result
s with a different approach and additional precision and generality.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Duke (UCLA)
DTSTART;VALUE=DATE-TIME:20221215T160000Z
DTEND;VALUE=DATE-TIME:20221215T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/162
DESCRIPTION:Title: On the analytic theory of isotropic ternary quadratic forms\nby
William Duke (UCLA) as part of Number Theory Web Seminar\n\n\nAbstract\nI
will describe recent work giving an asymptotic formula for a count of pri
mitive integral zeros of an isotropic ternary quadratic form in an orbit u
nder integral automorphs of the form. The constant in the asymptotic is ex
plicitly computed in terms of local data determined by the orbit. This is
compared with the well-known asymptotic for the count of all primitive
zeros. Together with an extension of results of Kneser by R. Schulze-Pill
ot on the classes in a genus of representations\, this yields a formula f
or the number of orbits\, summed over a genus of forms\, in terms of th
e number of local orbits. For a certain special class of forms a simple ex
plicit formula is given for this number.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Terence Tao (UCLA)
DTSTART;VALUE=DATE-TIME:20230223T160000Z
DTEND;VALUE=DATE-TIME:20230223T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/163
DESCRIPTION:Title: Infinite Partial Sumsets in the Primes\nby Terence Tao (UCLA) a
s part of Number Theory Web Seminar\n\n\nAbstract\nIt is an open question
as to whether the prime numbers contain the sum $A+B$ of two infinite sets
of natural numbers $A$\, $B$ (although results of this type are known ass
uming the Hardy-Littlewood prime tuples conjecture). Using the Maynard si
eve and the Bergelson intersectivity lemma\, we show the weaker result tha
t there exist two infinite sequences $a_1 < a_2 < ...$ and $b_1 < b_2 < ..
.$ such that $a_i + b_j$ is prime for all $i < j$. Equivalently\, the pri
mes are not "translation-finite" in the sense of Ruppert. As an applicati
on of these methods we show that the orbit closure of the primes is uncoun
table.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kannan Soundararajan (Stanford University)
DTSTART;VALUE=DATE-TIME:20230406T150000Z
DTEND;VALUE=DATE-TIME:20230406T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/164
DESCRIPTION:Title: Covering integers using quadratic forms\nby Kannan Soundararaja
n (Stanford University) as part of Number Theory Web Seminar\n\n\nAbstract
\nHow large must $\\Delta$ be so that we can cover a substantial proportio
n of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ w
ith $d$ below $\\Delta$? Problems involving representations by binary qua
dratic forms have a long history\, going back to Fermat. The particular p
roblem mentioned here was recently considered by Hanson and Vaughan\, and
Y. Diao. In ongoing work with Ben Green\, we resolve this problem\, and i
dentify a sharp phase transition: If $\\Delta$ is below $(\\log X)^{\\log
2-\\epsilon}$ then zero percent of the integers below $X$ are represented
\, whereas if $\\Delta$ is above $(\\log X)^{\\log 2 +\\epsilon}$ then 100
percent of the integers below $X$ are represented.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Régis de la Bretèche (Institut de Mathématiques de Jussieu-Pari
s Rive Gauche)
DTSTART;VALUE=DATE-TIME:20230112T160000Z
DTEND;VALUE=DATE-TIME:20230112T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/165
DESCRIPTION:Title: Higher moments of primes in arithmetic progressions\nby Régis
de la Bretèche (Institut de Mathématiques de Jussieu-Paris Rive Gauche)
as part of Number Theory Web Seminar\n\n\nAbstract\nIn a joint work with
Daniel Fiorilli\, we develop a new method to prove lower bounds of some mo
ments related to the distribution of primes in arithmetic progressions. We
shall present main results and explain some aspects of the proofs. To pr
ove our results\, we assume GRH but we succeed to avoid linearly independe
nce on zeroes hypothesis.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cecile Dartyge (Institut Élie Cartan\, Université de Lorraine)
DTSTART;VALUE=DATE-TIME:20230119T160000Z
DTEND;VALUE=DATE-TIME:20230119T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/166
DESCRIPTION:Title: On the largest prime factor of quartic polynomial values: the cycli
c and dihedral cases\nby Cecile Dartyge (Institut Élie Cartan\, Univ
ersité de Lorraine) as part of Number Theory Web Seminar\n\n\nAbstract\nL
et $P(X)$ be a monic\, quartic\, irreducible polynomial of $\\Z[X]$ with c
yclic or dihedral Galois group. We prove that there exists $c_P >0$\, such
that for a positive proportion of integers $n$\, $P(n)$ has a prime facto
r bigger than $n^{1+c_P}$. This is a joint work with James Maynard.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Wilms (University of Basel)
DTSTART;VALUE=DATE-TIME:20230126T160000Z
DTEND;VALUE=DATE-TIME:20230126T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/167
DESCRIPTION:Title: On equidistribution in Arakelov theory\nby Robert Wilms (Univer
sity of Basel) as part of Number Theory Web Seminar\n\n\nAbstract\nAs a mo
tivating example of its own interest I will first discuss a new equidistri
bution result for the zero sets of integer polynomials. More precisely\, I
will give a condition such that the zero sets tends to equidistribute wit
h respect to the Fubini-Study measure and I will show that this condition
is generically satisfied in sets of polynomials of bounded Bombieri norm.
In the second part\, I will embed this example in a much more general fram
ework about the distribution of the divisors of small sections of arithmet
ically ample hermitian line bundles in Arakelov theory.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (University of Michigan)
DTSTART;VALUE=DATE-TIME:20230504T150000Z
DTEND;VALUE=DATE-TIME:20230504T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/168
DESCRIPTION:Title: Counting nilpotent extensions\nby Peter Koymans (University of
Michigan) as part of Number Theory Web Seminar\n\n\nAbstract\nWe discuss s
ome recent progress towards the strong form of Malle’s conjecture. Even
for nilpotent extensions\, only very few cases of this conjecture are curr
ently known. We show how equidistribution of Frobenius elements plays an e
ssential role in this problem and how this can be used to make further pro
gress towards Malle’s conjecture. We will also discuss applications to t
he Massey vanishing conjecture and to lifting problems. This is joint work
with Carlo Pagano.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (University of the Witwatersrand)
DTSTART;VALUE=DATE-TIME:20230316T160000Z
DTEND;VALUE=DATE-TIME:20230316T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/169
DESCRIPTION:Title: Recent progress on the Skolem problem\nby Florian Luca (Univers
ity of the Witwatersrand) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nThe celebrated Skolem-Mahler-Lech Theorem states that the set of zeros
of a linear recurrence sequence is the union of a finite set and finitely
many arithmetic progressions. The corresponding computational question\,
the Skolem Problem\, asks to determine whether a given linear recurrence s
equence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost
90 years old\, decidability of the Skolem Problem remains open. One of th
e main contributions of the talk is to present an algorithm to solve the S
kolem Problem for simple linear recurrence sequences (those with simple ch
aracteristic roots). Whenever the algorithm terminates\, it produces a sta
nd-alone certificate that its output is correct -- a set of zeros together
with a collection of witnesses that no further zeros exist. We give a pro
of that the algorithm always terminates assuming two classical number-theo
retic conjectures: the Skolem Conjecture (also known as the Exponential Lo
cal-Global Principle) and the $p$-adic Schanuel Conjecture. Preliminary ex
periments with an implementation of this algorithm within the tool SKOLEM
point to the practical applicability of this method. \n In the second part
of the talk\, we present the notion of an Universal Skolem Set\, which is
a subset of the positive integers on which the Skolem is decidable regard
less of the linear recurrence. We give two examples of such sets\, one of
which is of positive density (that is\, contains a positive proportion of
all the positive integers).\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalín (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20230202T160000Z
DTEND;VALUE=DATE-TIME:20230202T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/170
DESCRIPTION:Title: Distributions of sums of the divisor function over function fields<
/a>\nby Matilde Lalín (Université de Montréal) as part of Number Theory
Web Seminar\n\n\nAbstract\nIn 2018 Keating\, Rodgers\, Roditty-Gershon an
d Rudnick studied the mean-square of sums of the divisor function $d_k(f)$
over short intervals and over arithmetic progressions for the function f
ield $\\mathbb{F}_q[T]$. By results from the Katz and Sarnak philosophy\,
they were able to relate these problems to certain integrals over the ens
emble of unitary matrices when $q$ goes to infinity. We study similar pro
blems leading to integrals over the ensembles of symplectic and orthogonal
matrices when $q$ goes to infinity. This is joint work with Vivian Kuperb
erg.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (Institute for Advanced Study and Princeton Universit
y)
DTSTART;VALUE=DATE-TIME:20230209T160000Z
DTEND;VALUE=DATE-TIME:20230209T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/171
DESCRIPTION:Title: Divisibility of character values of the symmetric group\nby Sar
ah Peluse (Institute for Advanced Study and Princeton University) as part
of Number Theory Web Seminar\n\n\nAbstract\nIn 2017\, Miller computed the
character tables of $S_n$ for all $n$ up to $38$ and looked at various sta
tistical properties of the entries. Characters of symmetric groups take on
ly integer values\, and\, based on his computations\, Miller conjectured t
hat almost all entries of the character table of $S_n$ are divisible by an
y fixed prime power as $n$ tends to infinity. In this talk\, I will discus
s joint work with K. Soundararajan that resolves this conjecture\, and men
tion some related open problems.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (University of Oxford)
DTSTART;VALUE=DATE-TIME:20230323T160000Z
DTEND;VALUE=DATE-TIME:20230323T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/172
DESCRIPTION:Title: How many smooth numbers and smooth polynomials are there?\nby O
fir Gorodetsky (University of Oxford) as part of Number Theory Web Seminar
\n\n\nAbstract\nSmooth numbers are integers whose prime factors are smalle
r than a threshold $y$. In the 80s they became important outside of pure m
ath\, as Pomerance's quadratic sieve for factoring integers relied on thei
r distribution. The density of smooth numbers up to $x$ can be approximate
d\, in some range\, using a peculiar function $\\rho$ called Dickman's fun
ction\, defined via a delay-differential equation. All of the above is als
o true for smooth polynomials over finite fields.\n\nWe'll survey these to
pics and discuss recent results concerning the range of validity of the ap
proximation of the density of smooth numbers by $\\rho$\, whose proofs rel
y on relating the counting function of smooth numbers to the Riemann zeta
function and the counting function of primes. In particular\, we uncover p
hase transitions in the behavior of the density at the points $y=(\\log x)
^2$ (as conjectured by Hildebrand) and $y=(\\log x)^(3/2)$\, when previous
ly only a transition at $y=\\log x$ was known and understood. These transi
tions also occur in the polynomial setting. We'll also show that a standar
d conjecture on the error in the Prime Number Theorem implies $\\rho$ is a
lways a lower bound for the density\, addressing a conjecture of Pomerance
.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao (Leibniz University Hannover)
DTSTART;VALUE=DATE-TIME:20230330T150000Z
DTEND;VALUE=DATE-TIME:20230330T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/173
DESCRIPTION:Title: Sparsity of rational and algebraic points\nby Ziyang Gao (Leibn
iz University Hannover) as part of Number Theory Web Seminar\n\n\nAbstract
\nIt is a fundamental question in mathematics to find rational solutions t
o a given system of polynomials\, and in modern language this question tra
nslates into finding rational points in algebraic varieties. This question
is already very deep for algebraic curves defined over $\\Q$. An intrinsi
c natural number associated with the curve\, called its genus\, plays an i
mportant role in studying the rational points on the curve. In 1983\, Falt
ings proved the famous Mordell Conjecture (proposed in 1922)\, which asser
ts that any curve of genus at least $2$ has only finitely many rational po
ints. Thus the problem for curves of genus at least $2$ can be divided int
o several grades: finiteness\, bound\, uniform bound\, effectiveness. An a
nswer to each grade requires a better understanding of the distribution of
the rational points.\n\nIn my talk\, I will explain the historical and re
cent developments of this problem according to the different grades.\n\nAn
other important topic on studying points on curves is the torsion packets.
This topic goes beyond rational points. I will also discuss briefly about
it in my talk.\n\nIf time permits\, I will mention the corresponding resu
lt in high dimensions.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20230216T160000Z
DTEND;VALUE=DATE-TIME:20230216T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/174
DESCRIPTION:Title: Ordinary and Basic Reductions of Abelian Varieties\nby Wanlin L
i (Université de Montréal) as part of Number Theory Web Seminar\n\n\nAbs
tract\nGiven an abelian variety A defined over a number field\, a conjectu
re attributed to Serre states that the set of primes at which A admits ord
inary reduction is of positive density. This conjecture had been proved fo
r elliptic curves (Serre\, 1977)\, abelian surfaces (Katz 1982\, Sawin 201
6) and certain higher dimensional abelian varieties (Pink 1983\, Fite 2021
\, etc). \n\nIn this talk\, we will discuss ideas behind these results and
recent progress for abelian varieties with non-trivial endomorphisms\, in
cluding some cases of A with almost complex multiplication by an abelian C
M field\, based on joint work with Cantoral-Farfan\, Mantovan\, Pries\, an
d Tang.\n\nApart from ordinary reduction\, we will also discuss the set of
primes at which an abelian variety admits basic reduction\, generalizing
a result of Elkies on the infinitude of supersingular primes for elliptic
curves. This is joint work with Mantovan\, Pries\, and Tang.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Le Boudec (University of Basel)
DTSTART;VALUE=DATE-TIME:20230427T150000Z
DTEND;VALUE=DATE-TIME:20230427T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/175
DESCRIPTION:Title: $2$-torsion in class groups of number fields\nby Pierre Le Boud
ec (University of Basel) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nIt is well-known that the class number of a number field $K$ of fixed d
egree $n$ is roughly bounded by the square root of the absolute value of t
he discriminant of $K$. However\, given a prime number $p$\, the cardinali
ty of the $p$-torsion subgroup of the class group of $K$ is expected to be
much smaller. Unfortunately\, beating the trivial bound mentioned above i
s a hard problem. Indeed\, this task had only been achieved for a handful
of pairs $(n\,p)$ until Bhargava\, Shankar\, Taniguchi\, Thorne\, Tsimerma
n and Zhao managed to do so for any degree $n$ in the case $p=2$. In this
talk we will go through their proof and we will present new bounds which d
epend on the geometry of the lattice underlying the ring of integers of $K
$. This is joint work with Dante Bonolis.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wilkie (University of Manchester)
DTSTART;VALUE=DATE-TIME:20230309T160000Z
DTEND;VALUE=DATE-TIME:20230309T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/176
DESCRIPTION:Title: Integer points on analytic sets\nby Alex Wilkie (University of
Manchester) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 2004 I
proved an $O(\\log\\log H)$ bound for the number of integer points of heig
ht at most $H$ lying on a globally subanalytic curve. (The paper was publi
shed in the Journal of Symbolic Logic and so probably escaped the notice o
f most of you reading this.) Recently\, Gareth Jones and Gal Binyamini pro
posed a generalization of the result to higher dimensions (where the obvio
us statement is almost certainly false) and I shall report on our joint wo
rk: one obtains the (hoped for) $(\\log\\log H)^n$ bound for (not globall
y subanalytic but) globally analytic sets of dimension $n$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Shlapentokh (East Carolina University)
DTSTART;VALUE=DATE-TIME:20230302T160000Z
DTEND;VALUE=DATE-TIME:20230302T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/177
DESCRIPTION:Title: Defining integers using unit groups\nby Alexandra Shlapentokh (
East Carolina University) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nWe discuss some problems of definability and decidability over rings o
f integers of algebraic extensions of $\\Q$. In particular\, we show that
for a large class of fields $K$ there is a simple formula defining ration
al integers over $O_K$. Below $U_K$ is the group of units of $O_K$. \n\n$
\\Z=\\{x| \\forall \\varepsilon \\in U_K\\setminus \\{1\\}\\ \\exists \\de
lta \\in U_K: x \\equiv \\frac{\\delta-1}{\\varepsilon-1} \\bmod (\\vareps
ilon-1)\\}$. This talk is based on a joint paper with Barry Mazur and Karl
Rubin.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20230420T130000Z
DTEND;VALUE=DATE-TIME:20230420T140000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/178
DESCRIPTION:Title: Integral points on curves via Baker's method and finite étale cove
rs\nby Bjorn Poonen (MIT) as part of Number Theory Web Seminar\n\n\nAb
stract\nWe prove results in the direction of showing that for some affine
curves\, Baker's method applied to finite étale covers is insufficient to
determine the integral points.\n\nPlease note the unusual time!\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary-Soroker (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20230601T150000Z
DTEND;VALUE=DATE-TIME:20230601T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/179
DESCRIPTION:Title: Random additive polynomials\nby Lior Bary-Soroker (Tel Aviv Uni
versity) as part of Number Theory Web Seminar\n\n\nAbstract\nRandom polyno
mials with integer coefficients tend to be irreducible and to have a large
Galois group with high probability. This was shown more than a century ag
o in the large box model\, where we choose the coefficients uniformly from
a box and let its size go to infinity\, while only recently there are res
ults in the restricted box model\, when the size of the box is bounded and
its dimension (i.e. the degree of the polynomial) goes to infinity. \n\nI
n this talk\, we will discuss an important class of random polynomials —
additive polynomials\, which have coefficients in the polynomial ring ove
r a finite field. In this case\, the roots form a vector space\, hence the
Galois group is naturally a subgroup of $\\GL_n$. \n\nWhile we prove that
the Galois group is the full matrix group both in the large box model\, a
nd in the large finite field limit\, our main result is in the restricted
box model: under some necessary condition the Galois group is large (in th
e sense that it contains $\\SL_n$) asymptotically almost surely\, as the d
egree goes to infinity.\n\nThe proof relies crucially on deep results on s
ubgroups of $\\GL_n$ by Fulman and Guralnick\, combined with tools from al
gebra and number theory. \n\nBased on a joint work with Alexei Entin and E
ilidh McKemmie\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Green (University of Oxford)
DTSTART;VALUE=DATE-TIME:20230511T150000Z
DTEND;VALUE=DATE-TIME:20230511T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/180
DESCRIPTION:Title: On Sarkozy's theorem for shifted primes\nby Ben Green (Universi
ty of Oxford) as part of Number Theory Web Seminar\n\n\nAbstract\nSuppose
that $N$ is large and that $A$ is a subset of $\\{1\,..\,N\\}$ which does
not contain two elements $x\, y$ with $x - y$ equal to $p-1$\, $p$ a prime
. Then $A$ has cardinality at most $N^{1 - c}$\, for some absolute $c > 0$
. I will discuss the history of this kind of question as well as some aspe
cts of the proof of the stated result.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/180/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART;VALUE=DATE-TIME:20230413T150000Z
DTEND;VALUE=DATE-TIME:20230413T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/181
DESCRIPTION:Title: Integrality Properties of the Betti Moduli Space\nby Hélène E
snault (Freie Universität Berlin) as part of Number Theory Web Seminar\n\
n\nAbstract\nWe use de Jong’s conjecture and the existence of $\\ell$-ad
ic companions to single out integrality properties of the Betti moduli spa
ce. The first such instance was in joint work with Michael Groechenig on S
impson’s integrality conjecture for (cohomologically) rigid local system
s. This integrality property yields an obstruction for a finitely presente
d group to be the fundamental group of a sooth quasi-projective complex va
riety. (joint with Johan de Jong)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/181/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard University)
DTSTART;VALUE=DATE-TIME:20230518T150000Z
DTEND;VALUE=DATE-TIME:20230518T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/182
DESCRIPTION:Title: Counting Minimally Ramified Global Field Extensions\nby Mark Sh
usterman (Harvard University) as part of Number Theory Web Seminar\n\n\nAb
stract\nGiven a finite group $G$\, one is interested in the number of Galo
is extensions of a global field with Galois group $G$ and bounded discrimi
nant. We consider a refinement of this problem where the discriminant is r
equired to have the smallest possible number of (distinct) prime factors.
We will discuss existing results and conjectures over number fields\, and
present some recent results over function fields.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barak Weiss (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20230525T150000Z
DTEND;VALUE=DATE-TIME:20230525T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/183
DESCRIPTION:Title: New bounds on lattice covering volumes\, and nearly uniform covers<
/a>\nby Barak Weiss (Tel Aviv University) as part of Number Theory Web Sem
inar\n\n\nAbstract\nLet $L$ be a lattice in $\\R^n$ and let $K$ be a conve
x body. The covering volume of $L$ with respect to $K$ is the minimal volu
me of a dilate $rK$\, such that $L+rK = \\R^n$\, normalized by the covolum
e of $L$. Pairs $(L\,K)$ with small covering volume correspond to efficien
t coverings of space by translates of $K$\, where the translates lie in a
lattice. Finding upper bounds on the covering volume as the dimension $n$
grows is a well studied problem in the so-called “Geometry of Numbers”
\, with connections to practical questions arising in computer science and
electrical engineering. In a recent paper with Or Ordentlich (EE\, Hebrew
University) and Oded Regev (CS\, NYU) we obtain substantial improvements
to bounds of Rogers from the 1950s. In another recent paper\, we obtain bo
unds on the minimal volume of nearly uniform covers (to be defined in the
talk). The key to these results are recent breakthroughs by Dvir and other
s regarding the discrete Kakeya problem. I will give an overview of the qu
estions and results.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/183/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART;VALUE=DATE-TIME:20230608T150000Z
DTEND;VALUE=DATE-TIME:20230608T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/184
DESCRIPTION:Title: On Chowla's non-vanishing conjecture over function fields\nby C
arlo Pagano (Concordia University) as part of Number Theory Web Seminar\n\
n\nAbstract\nA conjecture of Chowla postulates that no $L$-function of Dir
ichlet characters over the rationals vanishes at $s=1/2$. Soundararajan ha
s proved non-vanishing for a positive proportion of quadratic characters.
Over function fields Li has discovered that Chowla's conjecture fails for
infinitely many distinct quadratic characters. However\, on the basis of t
he Katz--Sarnak heuristics\, it is still widely believed that one should h
ave non-vanishing for 100% of the characters in natural families (such as
the family of quadratic characters). Works of Bui--Florea\, David--Florea-
-Lalin\, Ellenberg--Li--Shusterman\, among others\, provided evidence givi
ng a positive proportion of non-vanishing in several such families. I will
present an upcoming joint work with Peter Koymans and Mark Shusterman\, w
here we prove that for each fixed q congruent to $3$ modulo $4$ one has 10
0% non-vanishing in the family of imaginary quadratic function fields.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/184/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youness Lamzouri (Institut Elie Cartan de Lorraine)
DTSTART;VALUE=DATE-TIME:20230615T150000Z
DTEND;VALUE=DATE-TIME:20230615T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/185
DESCRIPTION:Title: A walk on Legendre paths\nby Youness Lamzouri (Institut Elie Ca
rtan de Lorraine) as part of Number Theory Web Seminar\n\n\nAbstract\nIn t
his talk\, we shall explore certain polygonal paths\, that we call ''Legen
dre paths''\, which encode important information about the values of the L
egendre symbol. More precisely\, the Legendre path modulo a prime number $
p$ is defined as the polygonal path in the plane whose vertices are the po
ints $(j\, S_p(j))$ for $0≤j≤p-1$\, where $S_p(j)$ is the (normalized)
sum of Legendre symbols $(n/p)$ for $n$ up to $j$. In particular\, we wi
ll attempt to answer the following questions as we vary over the primes $p
$: how are these paths distributed? how do their maximums behave? when do
es a Legendre path decreases for the first time? what is the typical numbe
r of $x$-intercepts of such paths? and what proportion of a Legendre path
is above the real axis? We will see that some of these questions correspo
nd to important and longstanding problems in analytic number theory\, incl
uding understanding the size of the least quadratic non-residue\, and impr
oving the Pólya-Vinogradov inequality for character sums. Among our resul
ts\, we prove that as we average over the primes\, the Legendre paths conv
erge in law\, in the space of continuous functions\, to a certain random F
ourier series constructed using Rademacher random multiplicative functions
. \n\nPart of this work is joint with Ayesha Hussain and with Oleksiy Klu
rman and Marc Munsch.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Εfthymios Sofos (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20230622T150000Z
DTEND;VALUE=DATE-TIME:20230622T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/186
DESCRIPTION:Title: The second moment method for rational points\nby Εfthymios Sof
os (University of Glasgow) as part of Number Theory Web Seminar\n\n\nAbstr
act\nIn a joint work with Alexei Skorobogatov we used a second-moment appr
oach to prove asymptotics for the average of the von Mangoldt function ove
r the values of a typical integer polynomial. As a consequence\, we proved
Schinzel's Hypothesis in 100% of the cases. In addition\, we proved that
a positive proportion of Châtelet equations have a rational point. I will
explain subsequent joint work with Tim Browning and Joni Teräväinen [ar
Xiv:2212.10373] that develops the method and establishes asymptotics for a
verages of an arithmetic function over the values of typical polynomials.
Part of the new ideas come from the theory of averages of arithmetic funct
ions in short intervals. One of the applications is that the Hasse princip
le holds for 100% of Châtelet equations. This agrees with the conjecture
of Colliot-Thélène stating that the Brauer--Manin obstruction is the onl
y obstruction to the Hasse principle for rationally connected varieties.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART;VALUE=DATE-TIME:20230629T150000Z
DTEND;VALUE=DATE-TIME:20230629T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/187
DESCRIPTION:Title: Some number theoretic aspects of finite point configurations\nb
y Alex Iosevich (University of Rochester) as part of Number Theory Web Sem
inar\n\n\nAbstract\nWe are going to survey some recent and less recent res
ults pertaining to the study of finite point configurations in Euclidean s
pace and vector spaces over finite fields\, centered around the Erdos/Falc
oner distance problems. We shall place particular emphasis on number-theor
etic ideas and obstructions that arise in this area.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/187/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (University of Wisconsin\, Madison)
DTSTART;VALUE=DATE-TIME:20231026T150000Z
DTEND;VALUE=DATE-TIME:20231026T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/188
DESCRIPTION:Title: Canonical heights on Shimura varieties and the Andre-Oort conjectur
e\nby Ananth Shankar (University of Wisconsin\, Madison) as part of Nu
mber Theory Web Seminar\n\n\nAbstract\nLet $S$ be a Shimura variety. The A
ndre-Oort conjecture posits that the Zariski closure of special points mus
t be a sub Shimura subvariety of $S$. The Andre-Oort conjecture for $A_g$
(the moduli space of principally polarized Abelian varieties) — and ther
efore its sub Shimura varieties — was proved by Jacob Tsimerman. However
\, this conjecture was unknown for Shimura varieties without a moduli inte
rpretation. Binyamini-Schmidt-Yafaev build on work of Binyamini to reduce
the Andre-Oort conjecture to establishing height bounds on special points.
I will describe joint work with Jonathan Pila and Jacob Tsimerman where w
e establish these height bounds\, and therefore prove the Andre Oort conje
cture in full generality.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/188/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Mangerel (Durham University)
DTSTART;VALUE=DATE-TIME:20230907T150000Z
DTEND;VALUE=DATE-TIME:20230907T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/189
DESCRIPTION:Title: Correlations\, sign patterns and rigidity theorems for multiplicati
ve functions\nby Alexander Mangerel (Durham University) as part of Num
ber Theory Web Seminar\n\n\nAbstract\nThe Liouville function $\\lambda(n)$
\, defined to be $+1$ for $n$ having an even number of prime factors (coun
ted with multiplicity) and $-1$ otherwise\, is a multiplicative function w
ith deep connections to the distribution of primes. Inspired by the prime
$k$-tuples conjecture of Hardy and Littlewood\, Chowla conjectured that fo
r every $k$ each of the $2^k$ distinct sign patterns\, i.e.\, tuples in $\
\{-1\,+1\\}^k$ are assumed by the tuples $(\\lambda(n+1)\,...\,\\lambda(n+
k))$\, $n \\in \\mathbb{N}$\, with the same asymptotic frequency.\n\nThe u
nderlying phenomenon at hand is that the prime factorisations of $n+1\,\\l
dots\,n+k$ are expected to be (in a precise sense) statistically independe
nt as $n$ varies. As conjectured by Elliott\, the same equidistribution of
sign patterns is expected to hold for other $\\pm 1$-valued multiplicativ
e functions\, provided they are ``far from being periodic''. To the best o
f our knowledge\, until recently no explicit constructions of multiplicati
ve functions with this behaviour were known. \n\nIn this talk we will dis
cuss precisely what Chowla's and Elliott's conjectures say\, survey some o
f the literature on correlations\, and discuss some related problems about
sign patterns. Specifically\, we will address:\n\ni) the construction of
``Liouville-like'' functions $f: \\mathbb{N} \\rightarrow \\{-1\,+1\\}$ wh
ose $k$-tuples $(f(n+1)\,...\,f(n+k))$ equidistribute in $\\{-1\,+1\\}^k$\
, answering a question of de la Rue from 2018\, and\n\nii) in the case $k
= 4$\, the classification of all $\\pm 1$-valued completely multiplicative
functions $f$ with the (rigid) property that the sequence of tuples $(f(n
+1)\,f(n+2)\,f(n+3)\,f(n+4))$ omits the pattern $(+1\,+1\,+1\,+1)$\, solvi
ng a 50-year old problem of R.H. Hudson.\n\nKey to these developments is a
new result about the vanishing of correlations of ``moderately aperiodic'
' multiplicative functions along a dense sequence of scales.\n\nBased on j
oint work with O. Klurman and J. Teräväinen.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/189/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20230914T160000Z
DTEND;VALUE=DATE-TIME:20230914T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/190
DESCRIPTION:Title: Murmurations of arithmetic $L$-functions\nby Andrew Sutherland
(MIT) as part of Number Theory Web Seminar\n\n\nAbstract\nWhile conducting
a series of number-theoretic machine learning experiments last year\, He\
, Lee\, Oliver\, and Pozdnyakov noticed a curious oscillation in the avera
ges of Frobenius traces of elliptic curves over $\\Q$. If one computes th
e average value of $a_p(E)$ for $E/\\Q$ of fixed rank with conductor in a
short interval\, as $p$ increases the average oscillates with a decaying f
requency determined by the conductor. That the rank influences the distri
bution of Frobenius traces has long been known (indeed\, this was the impe
tus for the experiments that led to the conjecture of Birch and Swinnerton
-Dyer)\, but these oscillations do not appear to have been noticed previou
sly. This may be due in part to the critical role played by the conductor
\; in arithmetic statistics it is common to order elliptic curves $E/\\Q$
by naive height rather than conductor\, but doing so obscures these oscill
ations.\n\nI will present results from an ongoing investigation of this ph
enomenon\, which is remarkably robust and not specific to elliptic curves.
One finds similar oscillations in the averages of Dirichlet coefficients
of many types of $L$-functions when organized by conductor and root number
\, including those associated to modular forms and abelian varieties. The
source of these murmurations in the case of weight-$2$ newforms with triv
ial nebentypus is now understood\, thanks to recent work of Zubrilina\, bu
t all other cases remain open.\n\nThis is based on joint work with Yang-Hu
i He\, Kyu-Hwan Lee\, Thomas Oliver\, and Alexey Pozdnyakov.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/190/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timothy Browning (IST Austria)
DTSTART;VALUE=DATE-TIME:20231019T150000Z
DTEND;VALUE=DATE-TIME:20231019T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/191
DESCRIPTION:Title: When is a random Diophantine equation soluble over $\\mathbb{Q}_p$
for all $p$?\nby Timothy Browning (IST Austria) as part of Number Theo
ry Web Seminar\n\n\nAbstract\nThe question in the title is of growing impo
rtance in number theory and represents a more tractable staging post than
the question of solubility over $\\mathbb{Q}$. \nI'll describe the landsca
pe for various families of varieties\, which can be interpreted as a more
delicate version of Manin's conjecture\, in which one counts rational poin
ts of bounded height which lie in the image of adelic points under a morp
hism. This leads to more subtle asymptotic behaviours and depends intimate
ly on the geometry of the morphism. This is joint work with Julian Lyczak\
, Roman Sarapin and Arne Smeets.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/191/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph H. Silverman (Brown University)
DTSTART;VALUE=DATE-TIME:20231102T150000Z
DTEND;VALUE=DATE-TIME:20231102T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/192
DESCRIPTION:Title: Field of Moduli and Fields of Definition in Arithmetic Geometry and
Arithmetic Dynamics\nby Joseph H. Silverman (Brown University) as par
t of Number Theory Web Seminar\n\n\nAbstract\nLet $X/\\overline{\\Q}$ be a
n algebraic variety defined over the field of algebraic numbers. We say th
at a number field $K$ is a field of definition (FOD) for $X$ if there is a
variety $Y/K$ such that $Y$ is $\\overline{\\Q}$-isomorphic to $X.$\n\nTh
e field of moduli (FOM) of $X$ is the fixed field of\n$$\n \\{ s
\\in G_\\Q : s(X) \\textrm{ is $\\overline{\\Q}$-isomorphic to $X$}\\}.\n$
$\nIt is easy to check that every FOD for $X$ contains the FOM of $X$\, bu
t there are many situations where the FOM of $X$ is not a FOD. I will brie
fly discuss the FOM versus FOD problem in the classical case of abelian va
rieties\, and then turn to the the analogous question for morphisms $f : \
\mathbb{P}^N \\longrightarrow \\mathbb{P}^N$ defined over $\\overline{\\Q}
$\, where two maps are (dynamically) isomorphic if they are conjugate by a
linear fractional transformation. I will describe what is known for $N=1$
\, including examples of maps for which the FOM is not an FOD. I will then
discuss recent results for higher dimensional projective spaces in which
we show that every map f has a FOD whose degree over its FOM is bounded by
a function depending only on $N$ and $\\deg(f)$. (Joint work with John D
oyle.)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/192/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (University of Bristol)
DTSTART;VALUE=DATE-TIME:20231109T160000Z
DTEND;VALUE=DATE-TIME:20231109T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/193
DESCRIPTION:Title: Smallest denominators\nby Jens Marklof (University of Bristol)
as part of Number Theory Web Seminar\n\n\nAbstract\nIf we partition the un
it interval into $3000$ equal subintervals and take the smallest denominat
or amongst all rational points in each subinterval\, what can we say about
the distribution of those $3000$ denominators? I will discuss this and re
lated questions\, its connection with Farey statistics and random lattices
. In particular\, I will report on higher dimensional versions of a recent
proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Martin [Bul
l. Sci. Math. 187 (2023)\, Paper No. 103305] on the convergence of the exp
ectation value of the above distribution\, as well as closely related work
by Chen and Haynes [Int. J. Number Theory 19 (2023)\, 1405--1413]. In fac
t\, we will uncover the full distribution and prove convergence of more mo
ments than just the expectation value. (This I believe was previously not
known even in one dimension.) We furthermore obtain a higher dimensional
extension of Kargaev and Zhigljavsky's work on moments of the distance fun
ction for the Farey sequence [J. Number Theory 65 (1997)\, 130--149] as we
ll as new results on pigeonhole statistics.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/193/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill University)
DTSTART;VALUE=DATE-TIME:20231116T160000Z
DTEND;VALUE=DATE-TIME:20231116T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/194
DESCRIPTION:Title: Explicit class field theory and orthogonal groups\nby Henri Dar
mon (McGill University) as part of Number Theory Web Seminar\n\n\nAbstract
\nEssentially all abelian extensions of the rational numbers or of a quadr
atic imaginary field\ncan be generated by special values of the exponentia
l function or of the modular $j$-function\nat explicit arguments in the g
round field. Describing the mathematical objects which could play the role
of trigonometric and modular functions in generating class fields of more
general base fields is the stated goal of explicit class field theory. Ar
ound 5 years ago Jan Vonk and I proposed a framework in which class field
s of real quadratic fields can be generated from the special values of ce
rtain “rigid meromorphic cocycles” at real quadratic arguments. Withou
t delving into the details of this framework\, I will present some simple
concrete consequences of it in settings where the base field is totally re
al\, and explain how they can be proved. The more general statements rest
on (but do not require the full force of)\nthe notion of rigid meromorphic
cocycles for orthogonal groups of signature $(r\,r)$ described in joint w
ork with Lennart Gehrmann and Mike Lipnowski\, and are also inspired by t
he calculations in Romain Branchereau’s PhD thesis. (Joint with Jan Von
k)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/194/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART;VALUE=DATE-TIME:20231005T150000Z
DTEND;VALUE=DATE-TIME:20231005T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/195
DESCRIPTION:Title: Brauer--Manin obstructions requiring arbitrarily many Brauer classe
s\nby Isabel Vogt (Brown University) as part of Number Theory Web Semi
nar\n\n\nAbstract\nA fundamental problem in the arithmetic of varieties ov
er global fields is to determine whether they have a rational point. As a
first effective step\, one can check that a variety has local points for
each place. However\, this is not enough\, as many classes of varieties a
re known to fail this local-global principle. The Brauer–Manin obstruct
ion to the local-global principle for rational points is captured by eleme
nts of the Brauer group. On a projective variety\, any Brauer–Manin obst
ruction is captured by a finite subgroup of the Brauer group. I will expl
ain joint work that shows that this subgroup can require arbitrarily many
generators. This is joint with J. Berg\, C. Pagano\, B. Poonen\, M. Stoll
\, N. Triantafillou and B. Viray.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/195/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20231130T160000Z
DTEND;VALUE=DATE-TIME:20231130T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/196
DESCRIPTION:Title: Diagonal cycles: some results and conjectures\nby Wei Zhang (MI
T) as part of Number Theory Web Seminar\n\n\nAbstract\nAlgebraic cycles ar
e among the most fundamental mathematical objects. I will discuss a class
of special algebraic cycles related to the diagonal cycle\, including the
Gross-Schoen cycle (the small diagonal) on the triple product of a curve\,
the arithmetic diagonal cycle appearing in the Gan-Gross-Prasad conjectur
e\, as well as the Fourier-Jacobi cycle defined by Yifeng Liu.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/196/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20230921T150000Z
DTEND;VALUE=DATE-TIME:20230921T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/197
DESCRIPTION:Title: Detecting primes in multiplicatively structured sequences\nby K
aisa Matomäki (University of Turku) as part of Number Theory Web Seminar\
n\n\nAbstract\nI will discuss a new sieve set-up which allows one to find
prime numbers in sequences that have a suitable multiplicative structure a
nd good "type I information". Among other things\, the method gives a new
L-function free proof of Linnik's theorem concerning the least prime in an
arithmetic progression. The talk is based on on-going joint work with Jor
i Merikoski and Joni Teräväinen.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/197/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Krieger (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20230928T150000Z
DTEND;VALUE=DATE-TIME:20230928T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/198
DESCRIPTION:Title: A transcendental birational dynamical degree\nby Holly Krieger
(University of Cambridge) as part of Number Theory Web Seminar\n\n\nAbstra
ct\nIn the study of a discrete dynamical system defined by polynomials\, w
e wish to understand the integer sequence formed by the degrees of the ite
rates of the map: examples of such a sequence include the Fibonacci and ot
her integer linear recurrence sequences\, but not all examples satisfy a f
inite recurrence. The growth of this sequence is measured by the dynamica
l degree\, an invariant which controls the topological\, arithmetic\, and
algebraic complexity of the system. I will discuss the surprising construc
tion\, joint with Bell\, Diller\, and Jonsson\, of a transcendental dynami
cal degree for a birational map of projective space\, and how our work fit
s into the general phenomenon of power series taking transcendental values
at algebraic inputs.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/198/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anke Pohl (University of Bremen)
DTSTART;VALUE=DATE-TIME:20231123T160000Z
DTEND;VALUE=DATE-TIME:20231123T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/199
DESCRIPTION:Title: Period functions for vector-valued automorphic functions via dynami
cs and cohomology\nby Anke Pohl (University of Bremen) as part of Numb
er Theory Web Seminar\n\n\nAbstract\nVector-valued automorphic functions\,
or generalized automorphic functions\, occur naturally in many areas\, mo
st notably in spectral theory\, number theory and mathematical physics. Al
ready Selberg promoted the idea to investigate vector-valued automorphic f
unctions alongside their classical relatives and to exploit their interact
ion in order to understand their properties. While during the last decades
the focus has been on automorphic functions equivariant with regard to un
itary representations\, the investigations recently turned to non-unitary
representations as well. I will report on the status of an ongoing project
to investigate simultaneously unitarily and non-unitarily equivariant aut
omorphic functions with a view towards period functions and a classical-qu
antum correspondence by means of dynamics (transfer operator methods) and
cohomology theory. This is joint work with R. Bruggeman.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/199/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Djordje Milićević (Bryn Mawr College)
DTSTART;VALUE=DATE-TIME:20231214T160000Z
DTEND;VALUE=DATE-TIME:20231214T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/200
DESCRIPTION:Title: Beyond the spherical sup-norm problem\nby Djordje Milićević (
Bryn Mawr College) as part of Number Theory Web Seminar\n\n\nAbstract\nThe
sup-norm problem on arithmetic Riemannian manifolds occupies a prominent
place at the intersection of harmonic analysis\, number theory\, and quant
um mechanics. It asks about the sup-norm of $L^2$-normalized joint eigenfu
nctions of invariant differential operators and Hecke operators — that i
s\, automorphic forms — most classically in terms of their Laplace eigen
values (as in the QUE problem for high-energy eigenstates)\, but also in t
erms of the volume of the manifold and other parameters.\n\nIn this talk\,
we will motivate the sup-norm problem and then describe our results\, joi
nt with Blomer\, Harcos\, and Maga\, which for the first time solve it for
non-spherical Maass forms of an increasing dimension of the associated $K
$-type\, on an arithmetic quotient of $G=\\SL(2\,\\C)$\, with $K=\\mathrm{
SU}(2)$. We combine representation theory\, spectral analysis\, and Diopha
ntine arguments\, developing new Paley-Wiener theory for $G$ and sharp est
imates on spherical trace functions of arbitrary $K$-type on the way to a
novel counting problem of Hecke correspondences close to various special s
ubmanifolds of $G$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/200/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Masser (University of Basel)
DTSTART;VALUE=DATE-TIME:20231012T150000Z
DTEND;VALUE=DATE-TIME:20231012T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/201
DESCRIPTION:Title: Some new elliptic integrals\nby David Masser (University of Bas
el) as part of Number Theory Web Seminar\n\n\nAbstract\nIn 1981 James Dave
nport surmised that if an algebraic function $f(x\,t)$ is not integrable (
with respect to $x$) by elementary means when $t$ is an independent variab
le\, then there are most finitely many complex numbers $\\tau$ such that $
f(x\,\\tau)$ is integrable by elementary means. Umberto Zannier and I in 2
020 obtained a couple of counterexamples and in broad principle classified
all of them with algebraic coefficients (they are necessarily somewhat ra
re). In this talk I will review our work\, describe our recent discovery o
f entire families of the things\, and sketch an indirect connexion with th
e counterexamples (known as Ribet curves) to ``relative Manin-Mumford'' fo
und by Daniel Bertrand in 2011.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/201/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Tucker (University of Rochester)
DTSTART;VALUE=DATE-TIME:20240125T160000Z
DTEND;VALUE=DATE-TIME:20240125T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/202
DESCRIPTION:Title: Tits and Borel type theorems for preperiodic points of finite morph
isms\nby Thomas Tucker (University of Rochester) as part of Number The
ory Web Seminar\n\n\nAbstract\nWe pose a general question: Given a finitel
y generated semigroup S of finite morphisms from a variety to itself\, wha
t can one say about how the structure of the semigroup is connected to the
relationship between the preperiodic points of the elements of S? When S
consists of polarized morphisms\, we can give a fairly simple answer to th
is question using Tate's limiting procedure for Weil and Moriwaki heights.
We formulate some conjectures that generalize this\nanswer and prove som
e results relating to these conjectures.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/202/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20231221T160000Z
DTEND;VALUE=DATE-TIME:20231221T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/203
DESCRIPTION:Title: A talk in honor of Peter Sarnak's 70th birthday\nby Zeev Rudnic
k (Tel Aviv University) as part of Number Theory Web Seminar\n\n\nAbstract
\nI will give selected highlights of Peter Sarnak's works on automorphic f
orms and some of the outstanding problems remaining.\n\nSpecial Chair: Ale
x Kontorovich (Rutgers University)\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/203/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Misha Rudnev (University of Bristol)
DTSTART;VALUE=DATE-TIME:20231207T160000Z
DTEND;VALUE=DATE-TIME:20231207T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/204
DESCRIPTION:Title: The sum-product problem for integers with few prime factors\nby
Misha Rudnev (University of Bristol) as part of Number Theory Web Seminar
\n\n\nAbstract\nIt was asked by Szemerédi if the known sum-product estima
tes can be improved for a set of $N$ integers under the constraint that ea
ch integer has a small number of prime factors. We prove\, if the maximum
number of prime factors for each integer is sub-logarithmic in $N$\, the s
um-product exponent $5/3-o(1)$. \n\nThis becomes a corollary of an additiv
e energy versus the product set cardinality estimate\, which turns out to
be the best possible. \nIt is based on a scheme of Burkholder-Gundy-Davis
martingale square function inequalities in $p$-adic scales\, followed by a
n application of a variant of the Schmidt subspace theorem.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/204/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oded Regev (Courant Institute of Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20240118T160000Z
DTEND;VALUE=DATE-TIME:20240118T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/205
DESCRIPTION:Title: An Efficient Quantum Factoring Algorithm\nby Oded Regev (Couran
t Institute of Mathematical Sciences) as part of Number Theory Web Seminar
\n\n\nAbstract\nWe show that n-bit integers can be factorized by independe
ntly running a quantum circuit with $\\tilde{O}(n^{3/2})$ gates for $\\sqr
t{n}+4$ times\, and then using polynomial-time classical post-processing.
In contrast\, Shor's algorithm requires circuits with $\\tilde{O}(n^2)$ ga
tes. The\ncorrectness of our algorithm relies on a number-theoretic heuris
tic assumption reminiscent of those used in subexponential classical facto
rization algorithms. It is currently not clear if the algorithm can lead t
o improved physical implementations in practice.\n\nNo background in quant
um computation will be assumed.\n\nBased on the arXiv preprint: https://ar
xiv.org/abs/2308.06572\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/205/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Y. Wang (IST Austria)
DTSTART;VALUE=DATE-TIME:20240208T160000Z
DTEND;VALUE=DATE-TIME:20240208T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/206
DESCRIPTION:Title: Sums of three cubes over a function field\nby Victor Y. Wang (I
ST Austria) as part of Number Theory Web Seminar\n\n\nAbstract\nI will tal
k about joint work with Tim Browning and Jakob Glas on producing sums of t
hree cubes over a function field\, assuming a $q$-restricted form of the R
atios Conjecture for a geometric family of $L$-functions. If time permits\
, I may also discuss some recent developments in homological stability tha
t could help to resolve this $q$-restricted Ratios Conjecture.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/206/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART;VALUE=DATE-TIME:20240201T160000Z
DTEND;VALUE=DATE-TIME:20240201T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/207
DESCRIPTION:Title: Recent progress towards the sum–product conjecture and related pr
oblems\nby Akshat Mudgal (University of Oxford) as part of Number Theo
ry Web Seminar\n\n\nAbstract\nAn important open problem in combinatorial n
umber theory is the Erdös–Szemerédi sum–product conjecture\, which s
uggests that for any positive integers $s$\, $N$\, and for any set $A$ of
$N$ integers\, either there are many $s$-fold sums of the form $a_1 + …
+ a_s$ or there are many $s$-fold products of the form $a_1…a_s$. While
this remains wide open\, various generalisations of this problem have been
considered more recently\, including the question of finding large additi
ve and multiplicative Sidon sets in arbitrary sets of integers as well as
studying the so-called low energy decompositions.\n\nIn this talk\, I will
outline some recent progress towards the above questions\, as well as hig
hlight how these connect very naturally to other key conjectures in additi
ve combinatorics.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/207/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART;VALUE=DATE-TIME:20240215T160000Z
DTEND;VALUE=DATE-TIME:20240215T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/208
DESCRIPTION:Title: Density of rational points near manifolds\nby Damaris Schindler
(Goettingen University) as part of Number Theory Web Seminar\n\n\nAbstrac
t\nGiven a bounded submanifold $M$ in $\\R^n$\, how many rational points w
ith common bounded denominator are there in a small thickening of $M$? Und
er what conditions can we count them asymptotically as the size of the den
ominator goes to infinity? I will discuss some recent work in this directi
on and arithmetic applications such as Serre's dimension growth conjecture
as well as applications in Diophantine approximation. For this I'll focus
on joint work with Shuntaro Yamagishi\, as well as joint work with Rajula
Srivastava and Niclas Technau.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/208/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Pollack (University of Georgia)
DTSTART;VALUE=DATE-TIME:20240229T160000Z
DTEND;VALUE=DATE-TIME:20240229T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/209
DESCRIPTION:Title: Stretching\, the truth about nonunique factorization\nby Paul P
ollack (University of Georgia) as part of Number Theory Web Seminar\n\n\nA
bstract\nNumber theorists learn at their mother's knee that unique factori
zation fails in $\\Z[\\sqrt{-5}]$. Less well-known is that $\\Z[\\sqrt{-5}
]$ exhibits only a "half-failure" of unique factorization: while two facto
rizations into irreducibles of the same element need not agree up to unit
factors\, their lengths (number of factors) does always agree. This is a s
pecial case of a 1960 result of Leonard Carlitz. I will discuss offshoots
of Carlitz's theorem. Particular attention will be paid to certain questio
ns of Coykendall regarding "elasticity" of orders in quadratic number fiel
ds.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/209/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20240509T150000Z
DTEND;VALUE=DATE-TIME:20240509T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/210
DESCRIPTION:by Jacob Tsimerman (University of Toronto) as part of Number T
heory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/210/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART;VALUE=DATE-TIME:20240222T160000Z
DTEND;VALUE=DATE-TIME:20240222T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/211
DESCRIPTION:Title: Secondary terms in the first moment of the 2-Selmer groups of ellip
tic curves\nby Arul Shankar (University of Toronto) as part of Number
Theory Web Seminar\n\n\nAbstract\nA web of interrelated conjectures (due t
o work of Goldfeld\, Katz--Sarnak\, Poonen-Rains\, Bhargava--Kane--Lenstra
--Poonen--Rains) predict the distributions of ranks and Selmer groups of e
lliptic curves over $\\Q$. These conjectures predict that the average rank
of elliptic curves is $1/2$. Furthermore\, it is known (due to Bhargava a
nd myself) that the average size of the $2$-Selmer group of elliptic curve
s is $3$ (when the family of all elliptic curves is ordered by (naive) hei
ght). \n\nOn the computational side\, Balakrishnan\, Ho\, Kaplan\, Spicer\
, Stein\, and Weigand collect and analyze data on ranks\, $2$-Selmer group
s\, and other arithmetic invariants of elliptic curves\, when ordered by h
eight. Interestingly\, they find both a larger average rank as well as a s
maller average size of the $2$-Selmer group in the data. In this talk\, w
e will discuss joint work with Takashi Taniguchi\, in which we give a poss
ible theoretical explanation for deviation of the data on $2$-Selmer group
s from the predicted distribution\, namely\, the existence of a secondary
term.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/211/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Munsch (Jean Monnet University)
DTSTART;VALUE=DATE-TIME:20240314T160000Z
DTEND;VALUE=DATE-TIME:20240314T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/212
DESCRIPTION:Title: Two tales on quadratic character sums\nby Marc Munsch (Jean Mon
net University) as part of Number Theory Web Seminar\n\n\nAbstract\nIn thi
s talk\, we report progress on two questions on sums of real Dirichlet cha
racters. \n\nFirstly\, we discuss quantitative results about the number o
f sign changes in the partial sums of the real character \n$\\chi_D$. Our
method allows us to locate these changes on a very short initial interval
(which goes beyond the range in Vinogradov's conjecture for the least quad
ratic non-residue). The flexibility or our method allows us to deduce simi
lar results in the case \nof random multiplicative functions. \n\n These r
esults are related with the location of real zeros of Fekete polynomials $
F_D$\, namely the polynomials whose coefficients are the values of \nthe r
eal character $\\chi_D$. \n\nIn a second part\, we will consider exponent
ial sums $\\sum_{n\\le D} \\chi_D(n) e(n\\theta)$ (in other words Fekete p
olynomial on the unit circle).\nRecently Harper showed that the restricted
sum up to $H$ converges (after normalization) to the standard complex Gau
ssian \nwhen both $\\chi_D$ and $\\theta\\in [0\,1]$ are selected uniforml
y at random and $H$ is small enough. We prove that\nthe distribution of t
he values of Fekete polynomials on the unit circle is very different and i
s governed by an explicit limiting (non-Gaussian) random point process. As
an application\, we solve an open problem about the Mahler measure of $F_
p$ as $p \\rightarrow +\\infty$. \n\nThis is based on joint works with Ol
eksiy Klurman and Youness Lamzouri.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/212/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (Caltech)
DTSTART;VALUE=DATE-TIME:20240307T160000Z
DTEND;VALUE=DATE-TIME:20240307T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/213
DESCRIPTION:Title: The next case after Apéry on mixed Tate periods\nby Vesselin D
imitrov (Caltech) as part of Number Theory Web Seminar\n\n\nAbstract\nI wi
ll introduce a method\, joint with Frank Calegari and Yunqing Tang\, for p
roving linear independence results and effective bad approximability measu
res. It is an outgrowth of our previous joint work on the so-called "unbou
nded denominators conjecture\," which was in some sense an application of
transcendental number theory to modular forms theory\, with the key step b
eing to prove sufficiently sharp $\\mathbb{Q}(x)$-linear dimension bounds
on certain spaces of algebraic functions. This time\, we step into the wil
der realm of G-functions with infinite monodromy\, and devise holonomy bou
nds fine enough to prove the linear independence of two certain Dirichlet
L-function values\, a result that\, in the realm of mixed Tate periods\, c
an be considered as the next-simplest case after Apery's proof of the irra
tionality of $\\zeta(3)$ (excluding the cases that reduce to the Hermite--
Lindemann theorem or the Gelfond--Baker theorem on linear forms in logarit
hms). One key input turns out to be the classical Shidlovsky lemma on func
tional bad approximability\, the point Siegel missed for three decades to
complete his theory of algebraic relations among special values of E-funct
ions. \n\nThis is all a joint work with Frank Calegari and Yunqing Tang.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/213/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Javier Fresán (Sorbonne University)
DTSTART;VALUE=DATE-TIME:20240321T160000Z
DTEND;VALUE=DATE-TIME:20240321T170000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/214
DESCRIPTION:Title: E-functions and Geometry\nby Javier Fresán (Sorbonne Universit
y) as part of Number Theory Web Seminar\n\n\nAbstract\nE-functions are pow
er series which solve a differential equation and whose coefficients are a
lgebraic numbers that satisfy certain growth conditions of arithmetic natu
re. They were introduced in Siegel's 1929 memoir on the applications of di
ophantine approximation with the goal of generalising the Hermite--Lindema
nn--Weierstrass theorem about the transcendence of the values of the expon
ential function at algebraic arguments. Besides the exponential\, standard
examples include the Bessel function and confluent hypergeometric series.
After briefly surveying on the history of E-functions\, I will present a
joint work in progress with Peter Jossen where we prove that exponential p
eriod functions provide us with a rich geometric source of E-functions. Th
e easiest examples\, attached to polynomials of degree 4\, already allowed
us a couple of years ago to exhibit some E-functions which are not polyno
mial expressions in hypergeometric series\, thus solving one of the proble
ms in Siegel's original paper.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/214/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Princeton University)
DTSTART;VALUE=DATE-TIME:20240516T150000Z
DTEND;VALUE=DATE-TIME:20240516T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/215
DESCRIPTION:by Nina Zubrilina (Princeton University) as part of Number The
ory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/215/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Petersen (Stockholm University)
DTSTART;VALUE=DATE-TIME:20240404T150000Z
DTEND;VALUE=DATE-TIME:20240404T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/216
DESCRIPTION:Title: Moments of families of quadratic $L$-functions over function fields
via homotopy theory\nby Dan Petersen (Stockholm University) as part o
f Number Theory Web Seminar\n\n\nAbstract\nThis is a report of joint work
with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There
is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows
for precise predictions for the asymptotics of moments of many different
families of $L$-functions. Our work concerns the CFKRS predictions in the
case of the quadratic family over function fields\, i.e. the family of all
$L$-functions attached to hyperelliptic curves over some fixed finite fie
ld. One can relate this problem to understanding the homology of the braid
group with certain symplectic coefficients. With Bergström-Diaconu-Weste
rland we compute the stable homology groups of the braid groups with these
coefficients\, together with their structure as Galois representations. W
e moreover show that the answer matches the number-theoretic predictions.
With Miller-Patzt-Randal-Williams we prove a uniform range for homological
stability with these coefficients. Together\, these results imply the CFK
RS predictions for all moments in the function field case\, for all suffic
iently large (but fixed) $q$.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/216/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART;VALUE=DATE-TIME:20240523T150000Z
DTEND;VALUE=DATE-TIME:20240523T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/217
DESCRIPTION:by John Voight (Dartmouth College) as part of Number Theory We
b Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/217/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiannis Petridis (University College London)
DTSTART;VALUE=DATE-TIME:20240411T150000Z
DTEND;VALUE=DATE-TIME:20240411T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/218
DESCRIPTION:Title: Counting and equidistribution\nby Yiannis Petridis (University
College London) as part of Number Theory Web Seminar\n\n\nAbstract\nI will
discuss how counting orbits in hyperbolic spaces lead to interesting numb
er theoretic problems. The counting problems (and the associated equidistr
ibution) can be studied with various methods\, and I will emphasize automo
rphic form techniques\, originating in the work of H. Huber and studied ex
tensively by A. Good. My collaborators in various aspects of this project
are Chatzakos\, Lekkas\, Risager\, and Voskou.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/218/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theresa Anderson (Carnegie Mellon University)
DTSTART;VALUE=DATE-TIME:20240425T150000Z
DTEND;VALUE=DATE-TIME:20240425T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/219
DESCRIPTION:Title: Counting with new tools\nby Theresa Anderson (Carnegie Mellon U
niversity) as part of Number Theory Web Seminar\n\n\nAbstract\nArithmetic
statistics\, or the counting of objects of algebraic interest\, has seen a
lot of development in the last twenty years. We will take a glimpse into
just a few recent advances\, with an emphasis on the wide interplay of new
tools and techniques.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/219/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rainer Dietmann (Royal Holloway\, University of London)
DTSTART;VALUE=DATE-TIME:20240530T150000Z
DTEND;VALUE=DATE-TIME:20240530T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/220
DESCRIPTION:by Rainer Dietmann (Royal Holloway\, University of London) as
part of Number Theory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/220/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (EPFL)
DTSTART;VALUE=DATE-TIME:20240613T150000Z
DTEND;VALUE=DATE-TIME:20240613T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/221
DESCRIPTION:by Philippe Michel (EPFL) as part of Number Theory Web Seminar
\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/221/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wouter Castryck (KU Leuven)
DTSTART;VALUE=DATE-TIME:20240328T170000Z
DTEND;VALUE=DATE-TIME:20240328T180000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/222
DESCRIPTION:Title: The isogeny interpolation problem\nby Wouter Castryck (KU Leuve
n) as part of Number Theory Web Seminar\n\n\nAbstract\nIt is easy to prove
that a degree-$d$ isogeny $f$ between two elliptic curves $E$ and $E'$ is
completely determined by the images of any $4d + 1$ points. In this talk
we will study the algorithmic problem of evaluating $f$ at a given point $
P$ on $E$\, merely upon input of such "interpolation data". In case the in
terpolation points generate a group containing $E[N]$ such that $N^2 > 4d$
is smooth and coprime to $d$ and the field characteristic\, this problem
was solved in 2022 by Robert\, in the context of breaking SIKE (= SIDH)\,
a former candidate for post-quantum key exchange that had advanced to the
final stage of a standardization effort run by the National Institute of S
tandards and Technology. We will discuss this solution\, and then show how
to address more general instances of the isogeny interpolation problem\,
while also publicizing some unsolved cases.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/222/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uri Shapira (Technion – Israel Institute of Technology)
DTSTART;VALUE=DATE-TIME:20240418T150000Z
DTEND;VALUE=DATE-TIME:20240418T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/223
DESCRIPTION:Title: Distribution of conditional directional lattices\nby Uri Shapir
a (Technion – Israel Institute of Technology) as part of Number Theory W
eb Seminar\n\n\nAbstract\nGiven an integral vector $v$ in Euclidean $n$-sp
ace we project the standard lattice $\\Z^n$ into the hyperplane orthogonal
to $v$ and obtain in this manner a "lattice of rank $n-1$" in that hyperp
lane\, which is called "The directional lattice $D(\\Z^n\,v)$". \n\nIn thi
s talk I will discuss results about the limit distribution of directional
lattices as we let the vector $v$ vary in some natural sets from a number
theoretic point of view. These include\, balls\, spheres\, non-compact qua
dratic surfaces\, and integral vectors approximating an irrational line.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/223/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Daw (University of Reading)
DTSTART;VALUE=DATE-TIME:20240627T150000Z
DTEND;VALUE=DATE-TIME:20240627T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/224
DESCRIPTION:by Chris Daw (University of Reading) as part of Number Theory
Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/224/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin–Madison)
DTSTART;VALUE=DATE-TIME:20240926T150000Z
DTEND;VALUE=DATE-TIME:20240926T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/225
DESCRIPTION:by Jordan Ellenberg (University of Wisconsin–Madison) as par
t of Number Theory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/225/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dang-Khoa Nguyen (University of Calgary)
DTSTART;VALUE=DATE-TIME:20240606T150000Z
DTEND;VALUE=DATE-TIME:20240606T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/226
DESCRIPTION:by Dang-Khoa Nguyen (University of Calgary) as part of Number
Theory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/226/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Sah (MIT)
DTSTART;VALUE=DATE-TIME:20240502T150000Z
DTEND;VALUE=DATE-TIME:20240502T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/227
DESCRIPTION:Title: Quasipolynomial bounds on the inverse theorem for the Gowers norms
and applications\nby Ashwin Sah (MIT) as part of Number Theory Web Sem
inar\n\n\nAbstract\nRecent work\, joint with James Leng and Mehtaab Sawhne
y\, improves the so-called “inverse theorem” for the Gowers $U^{s+1}[N
]$-norm which arises in the field of additive combinatorics in relation to
Roth’s and Szemerédi’s theorems. I will explain how the field of hig
her-order Fourier analysis broadly extends Fourier methods and the circle
method in number theory\, and discuss implications of bounds for inverse t
heorems.\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/227/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Wenqiang Xu (Stanford University)
DTSTART;VALUE=DATE-TIME:20240620T150000Z
DTEND;VALUE=DATE-TIME:20240620T160000Z
DTSTAMP;VALUE=DATE-TIME:20240423T120329Z
UID:NTWebSeminar/228
DESCRIPTION:by Max Wenqiang Xu (Stanford University) as part of Number The
ory Web Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NTWebSeminar/228/
END:VEVENT
END:VCALENDAR