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BEGIN:VEVENT
SUMMARY:Emmanuel Breuillard (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20200918T161500Z
DTEND;VALUE=DATE-TIME:20200918T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/1
DESCRIPTION:Title: A
subspace theorem for manifolds\nby Emmanuel Breuillard (University of
Cambridge) as part of New England Dynamics and Number Theory Seminar\n\n\n
Abstract\nSchmidt’s subspace theorem is a fundamental result in diophant
ine approximation and a natural generalization of Roth’s celebrated theo
rem. In this talk I will discuss a geometric understanding of this theorem
that blends homogeneous dynamics and geometric invariant theory. Combined
with the Kleinbock-Margulis quantitative non-divergence estimates this yi
elds a natural generalization of the subspace theorem to systems of linear
forms that depend nicely on a parameter. I will also present several appl
ications and consequences of the main result. Joint work with Nicolas de S
axcé.\n
LOCATION:https://researchseminars.org/talk/NEDNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yotam Smilansky (Rutgers University)
DTSTART;VALUE=DATE-TIME:20200925T161500Z
DTEND;VALUE=DATE-TIME:20200925T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/2
DESCRIPTION:Title: Mu
ltiscale substitution tilings\nby Yotam Smilansky (Rutgers University)
as part of New England Dynamics and Number Theory Seminar\n\n\nAbstract\n
Multiscale substitution tilings are a new family of tilings of Euclidean s
pace that are generated by multiscale substitution rules. Unlike the stand
ard setup of substitution tilings\, which is a basic object of study withi
n the aperiodic order community and includes examples such as the Penrose
and the pinwheel tilings\, multiple distinct scaling constants are allowed
\, and the defining process of inflation and subdivision is a continuous o
ne. Under a certain irrationality assumption on the scaling constants\, th
is construction gives rise to a new class of tilings\, tiling spaces and t
iling dynamical systems\, which are intrinsically different from those tha
t arise in the standard setup. In the talk I will describe these new objec
ts and discuss various structural\, geometrical\, statistical and dynamica
l results. Based on joint work with Yaar Solomon.\n
LOCATION:https://researchseminars.org/talk/NEDNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samantha Fairchild (University of Washington)
DTSTART;VALUE=DATE-TIME:20201002T161500Z
DTEND;VALUE=DATE-TIME:20201002T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/3
DESCRIPTION:Title: Co
unting social interactions for discrete subsets of the plane\nby Saman
tha Fairchild (University of Washington) as part of New England Dynamics a
nd Number Theory Seminar\n\n\nAbstract\nGiven a discrete subset V in the p
lane\, how many points would you expect there to be in a ball of radius 10
0? What if the radius is 10\,000? Due to the results of Fairchild and fort
hcoming work with Burrin\, when V arises as orbits of non-uniform lattice
subgroups of SL(2\,R)\, we can understand asymptotic growth rate with erro
r terms of the number of points in V for a broad family of sets. A crucial
aspect of these arguments and similar arguments is understanding how to c
ount pairs of saddle connections with certain properties determining the i
nteractions between them\, like having a fixed determinant or having anoth
er point in V nearby. We will focus on a concrete case used to state the t
heorem and highlight the proof strategy. We will also discuss some ongoing
work and ideas which advertise the generality and strength of this argume
nt.\n
LOCATION:https://researchseminars.org/talk/NEDNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nattalie Tamam (University of California\, San Diego)
DTSTART;VALUE=DATE-TIME:20201009T161500Z
DTEND;VALUE=DATE-TIME:20201009T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/4
DESCRIPTION:Title: Ef
fective equidistribution of horospherical flows in infinite volume\nby
Nattalie Tamam (University of California\, San Diego) as part of New Engl
and Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstr
act\nWe want to provide effective information about averages of orbits of
the horospherical subgroup acting on a hyperbolic manifold of infinite vol
ume. We start by presenting the setting and results for manifolds with fin
ite volume. Then\, discuss the difficulties that arise when studying the i
nfinite volume setting\, and the measures that play a crucial role in it.
This is joint work with Jacqueline Warren.\n
LOCATION:https://researchseminars.org/talk/NEDNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Douglas Lind (University of Washington)
DTSTART;VALUE=DATE-TIME:20201016T161500Z
DTEND;VALUE=DATE-TIME:20201016T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/5
DESCRIPTION:Title: De
cimation limits of algebraic actions\nby Douglas Lind (University of W
ashington) as part of New England Dynamics and Number Theory Seminar\n\nLe
cture held in Online.\n\nAbstract\nThis is intended to be an expository ta
lk using simple examples to illustrate what’s going on\, and so will (ho
pefully) be a gentle introduction to these topics. Given a polynomial in d
commuting variables we can define an algebraic action of ℤ^d by commuti
ng automorphisms of a compact subgroup of 𝕋^(ℤ^d). Restricting the co
ordinates of points in this group to finite-index subgroups of ℤ^d gives
other algebraic actions\, defined by polynomials whose support grows poly
nomially and whose coefficients grow exponentially. But by “renormalizin
g” we can obtain a limiting object that is a concave function on ℝ^d w
ith interesting properties\, e.g. its maximum value is the entropy of the
action. For some polynomials this function also arises in statistical mech
anics models as the “surface tension” of a random surface via a variat
ional principle. In joint work with Arzhakova\, Schmidt\, and Verbitskiy\,
we establish this limiting behavior\, and identify the limit in terms of
the Legendre transform of the Ronkin function of the polynomial. The proof
is based on Mahler’s estimates on polynomial coefficients using Mahler
measure\, and an idea used by Boyd to prove that Mahler measure is continu
ous in the coefficients of the polynomial. Refinements of convergence ques
tions involve diophantine issues that I will discuss\, together with some
open problems.\n
LOCATION:https://researchseminars.org/talk/NEDNT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mishel Skenderi (Brandeis University)
DTSTART;VALUE=DATE-TIME:20201023T161500Z
DTEND;VALUE=DATE-TIME:20201023T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/6
DESCRIPTION:Title: Sm
all values at integer points of generic subhomogeneous functions\nby M
ishel Skenderi (Brandeis University) as part of New England Dynamics and N
umber Theory Seminar\n\nLecture held in Online.\n\nAbstract\nThis talk wil
l be based on joint work with Dmitry Kleinbock that has been motivated by
several recent papers (among them\, those of Athreya-Margulis\, Bourgain\,
Ghosh-Gorodnik-Nevo\, Kelmer-Yu). Given a certain sort of group $G$ and c
ertain sorts of functions $f: \\mathbb{R}^n \\to \\mathbb{R}$ and $\\psi :
\\mathbb{R}^n \\to \\mathbb{R}_{>0}\,$ we obtain necessary and sufficient
conditions so that for Haar-almost every $g \\in G\,$ there exist infinit
ely many (respectively\, finitely many) $v \\in \\mathbb{Z}^n$ for which $
|(f \\circ g)(v)| \\leq \\psi(\\|v\\|)\,$ where $\\|\\cdot\\|$ is an arbit
rary norm on $\\mathbb{R}^n.$ We also give a sufficient condition in the s
etting of uniform approximation. As a consequence of our methods\, we obta
in generalizations to the case of vector-valued (simultaneous) approximati
on with no additional effort. In our work\, we use probabilistic results i
n the geometry of numbers that go back several decades to the work of Sieg
el\, Rogers\, and W. Schmidt\; these results have recently found new life
thanks to a 2009 paper of Athreya-Margulis.\n
LOCATION:https://researchseminars.org/talk/NEDNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Sanchez (University of Washington)
DTSTART;VALUE=DATE-TIME:20201030T161500Z
DTEND;VALUE=DATE-TIME:20201030T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/7
DESCRIPTION:Title: Ga
ps of saddle connection directions for some branched covers of tori\nb
y Anthony Sanchez (University of Washington) as part of New England Dynami
cs and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nConsi
der the class of translation surfaces given by gluing two identical tori a
long a slit. Every such surface has genus two and two cone-type singularit
ies of angle $4\\pi$. There is a distinguished set of geodesics called sad
dle connections that are the geodesics between cone points. We can recover
a vector in the plane representing the saddle connection by keeping track
of the amount that the saddle connection moves in the vertical and horizo
ntal direction. How random is the set of saddle connections? \nWe motivate
the gap distribution of slopes as a measure of randomness and compute the
gap distribution of slopes of saddle connections for the class of transla
tion surfaces given by gluing two identical tori along a slit.\n
LOCATION:https://researchseminars.org/talk/NEDNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Byungchul Cha (Muhlenberg College)
DTSTART;VALUE=DATE-TIME:20201106T171500Z
DTEND;VALUE=DATE-TIME:20201106T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/8
DESCRIPTION:Title: In
trinsic Diophantine Approximation of circles\nby Byungchul Cha (Muhlen
berg College) as part of New England Dynamics and Number Theory Seminar\n\
nLecture held in Online.\n\nAbstract\nLet $S^1$ be the unit circle in $\\m
athbb{R}^2$ centered at the origin and let $Z$ be a countable dense subset
of $S^1$\, for instance\, the set $Z = S^1(\\mathbb{Q})$ of all rational
points in $S^1$. We give a complete description of an initial discrete par
t of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine
approximation. This is an analogue of the classical result of Markoff in 1
879\, where he characterized the most badly approximable real numbers via
the periods of their continued fraction expansions. Additionally\, we pres
ent similar results for a few different subsets $Z$ of $S^1$. This is join
t work with Dong Han Kim.\n
LOCATION:https://researchseminars.org/talk/NEDNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacqueline Warren (University of California\, San Diego)
DTSTART;VALUE=DATE-TIME:20201113T171500Z
DTEND;VALUE=DATE-TIME:20201113T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/9
DESCRIPTION:Title: Jo
ining classification and factor rigidity in infinite volume\nby Jacque
line Warren (University of California\, San Diego) as part of New England
Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\
nFor a group acting on two spaces\, a joining of these systems is a measur
e on the product space that is invariant under the diagonal action and pro
jects to the original measures on each space. As an important step towards
her celebrated measure classification theorem\, Ratner proved an early la
ndmark result classifying joinings for horocycle flows on finite volume qu
otients of PSL(2\,R). In this talk\, I will discuss joining classification
for horospherical flows in the infinite volume\, rank one setting\, as we
ll as a key factor rigidity theorem that is used in the proof.\n
LOCATION:https://researchseminars.org/talk/NEDNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahriar Mirzadeh (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201120T171500Z
DTEND;VALUE=DATE-TIME:20201120T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/10
DESCRIPTION:Title: O
n the dimension drop conjecture for diagonal flows on the space of lattice
s\nby Shahriar Mirzadeh (Michigan State University) as part of New Eng
land Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbst
ract\nConsider the set of points in a homogeneous space X=G/Gamma whose g_
t orbit misses a fixed open set. It has measure zero if the flow is ergodi
c. It has been conjectured that this set has Hausdorff dimension strictly
smaller than the dimension of X. This conjecture is proved when X is compa
ct or when it has real rank 1. In this talk we will prove the conjecture f
or probably the most important example of the higher rank case\, namely: G
=SL(m+n\, R)\, Gamma=SL(m+n\,Z)\, and g_t = diag(exp(t/m)\, …\, exp(t/m)
\, exp(-t/n)\, …\, exp(-t/n)). We can also use our main result to produc
e new applications to Diophantine approximation. This project is joint wor
k with Dmitry Kleinbock.\n
LOCATION:https://researchseminars.org/talk/NEDNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Sanchez (University of Washington)
DTSTART;VALUE=DATE-TIME:20201211T171500Z
DTEND;VALUE=DATE-TIME:20201211T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/12
DESCRIPTION:Title: G
aps of saddle connection directions for some branched covers of tori\n
by Anthony Sanchez (University of Washington) as part of New England Dynam
ics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nHolo
nomy vectors of translation surfaces provide a geometric generalization fo
r higher genus surfaces of (primitive) integer lattice points. The countin
g and distribution properties of holonomy vectors on translation surfaces
have been studied extensively. In this talk\, we consider the following qu
estion: How random are the holonomy vectors of a translation surface? We m
otivate the gap distribution of slopes of holonomy vectors as a measure of
randomness and compute the gap distribution for the class of translation
surfaces given by gluing two identical tori along a slit. No prior backgro
und on translation surfaces or gap distributions will be assumed.\n
LOCATION:https://researchseminars.org/talk/NEDNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Osama Khalil (University of Utah)
DTSTART;VALUE=DATE-TIME:20201204T171500Z
DTEND;VALUE=DATE-TIME:20201204T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/13
DESCRIPTION:Title: L
arge centralizers and counting integral points on affine varieties\nby
Osama Khalil (University of Utah) as part of New England Dynamics and Num
ber Theory Seminar\n\nLecture held in Online.\n\nAbstract\nDuke-Rudnick-Sa
rnak and Eskin-McMullen initiated the use of ergodic methods to count inte
gral points on affine homogeneous varieties. They reduced the problem to o
ne of studying limiting distributions of translates of periods of reductiv
e groups on homogeneous spaces. The breakthrough of Eskin\, Mozes and Shah
provided a rather complete understanding of this question in the case the
reductive group has a “small centralizer” inside the ambient group. I
n this talk\, we describe work in progress giving new results on the equid
istribution of generic translates of closed orbits of semisimple groups wi
th “large centralizers”. The key new ingredient is an algebraic descri
ption of a partial compactification (for lack of a better word) of the set
of intermediate groups which act as obstructions to equidistribution. Thi
s allows us to employ tools from geometric invariant theory to study the a
voidance problem.\n
LOCATION:https://researchseminars.org/talk/NEDNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cagri Sert (Universität Zürich)
DTSTART;VALUE=DATE-TIME:20210201T171500Z
DTEND;VALUE=DATE-TIME:20210201T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/14
DESCRIPTION:Title: E
xpanding measures and random walks on homogeneous spaces\nby Cagri Ser
t (Universität Zürich) as part of New England Dynamics and Number Theory
Seminar\n\nLecture held in Online.\n\nAbstract\nWe will start by reviewin
g some recent works on random walks on homogeneous spaces. We will continu
e by discussing the notion of a H-expanding probability measure on a conne
cted semisimple Lie group H\, that we introduce inspired by these developm
ents. As we shall see\, for a H-expanding µ with H < G\, on the one hand\
, one can obtain a description of µ-stationary probability measures on th
e homogeneous space G/Λ using the measure classification results of Eskin
– Lindenstrauss\, and on the other hand\, the recurrence techniques of B
enoist–Quint can be generalized to this setting. As a result\, we will d
educe equidistribution and orbit closure description results simultaneousl
y for a class of subgroups which contains Zariski-dense subgroups and some
epimorphic subgroups of H. If time allows\, we will see how\, using an id
ea of Simmons–Weiss\, this allows also us to deduce Birkhoff genericity
of a class of fractal measures with respect to expanding diagonal actions.
Joint work with Roland Prohaska and Ronggang Shi.\n
LOCATION:https://researchseminars.org/talk/NEDNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barak Weiss (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20210208T171500Z
DTEND;VALUE=DATE-TIME:20210208T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/15
DESCRIPTION:Title: C
lassification and statistics of cut-and-project sets\nby Barak Weiss (
Tel Aviv University) as part of New England Dynamics and Number Theory Sem
inar\n\nLecture held in Online.\n\nAbstract\nWe introduce a class of so-ca
lled “Ratner-Marklof-Strombergsson measures”. These are probability me
asures supported on cut-and-project sets in Euclidean space of dimension d
>1 which are invariant and ergodic for the action of the groups ASL_d(R) o
r SL_d(R) (affine or linear maps preserving orientation and volume). We cl
assify the measures that can arise in terms of algebraic groups and homoge
neous dynamics. Using the classification\, we prove analogues of results o
f Siegel\, Weil and Rogers about a Siegel summation formula and identities
and bounds involving higher moments. We deduce results about asymptotics\
, with error estimates\, of point-counting and patch-counting for typical
cut-and-project sets. Joint work with Rene Ruehr and Yotam Smilansky.\n
LOCATION:https://researchseminars.org/talk/NEDNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tsviqa Lakrec (The Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20210222T171500Z
DTEND;VALUE=DATE-TIME:20210222T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/16
DESCRIPTION:Title: E
quidistribution of affine random walks on some nilmanifolds\nby Tsviqa
Lakrec (The Hebrew University of Jerusalem) as part of New England Dynami
cs and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nWe co
nsider the action of the group of affine transformations on a nilmanifold.
Given a probability measure on this group and a starting point\, a random
walk on the nilmanifold is defined. We study quantitative equidistributio
n in law of such affine random walks on nilmanifolds. Under certain assump
tions\, we show that a failure to have fast equidistribution on a nilmanif
old is due to a failure on some factor nilmanifold. Combined with equidist
ribution results on the torus\, this leads to an equidistribution statemen
t on some nilmanifolds\, such as Heisenberg nilmanifolds.\nThis talk is ba
sed on joint works with Weikun He and Elon Lindenstrauss.\n
LOCATION:https://researchseminars.org/talk/NEDNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Kirsebom (University of Hamburg)
DTSTART;VALUE=DATE-TIME:20210315T161500Z
DTEND;VALUE=DATE-TIME:20210315T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/17
DESCRIPTION:Title: T
owards an extreme value law for the deepest cusp excursions of the unipote
nt flow\nby Maxim Kirsebom (University of Hamburg) as part of New Engl
and Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstr
act\nThe unipotent flow on the unit tangent bundle of the modular surface
is a classic example of a homogeneous flow when understood through the ide
ntification with PSL_2(R)/PSL_2(Z). The ergodicity of the flow implies tha
t almost every orbit is dense in the space and hence must eventually make
excursions deeper and deeper into the cusp. We are interested in understan
ding the nature of these excursions. In the described setting\, and more g
enerally\, Athreya and Margulis proved that the maximal excursions obey th
e logarithm law almost surely\, meaning that their growth rate scales the
logarithm of the time. In this work we focus on a more precise description
of this behaviour\, namely determining the probability that the deepest e
xcursion fails to outperform the expected asymptotic behaviour by an addit
ive amount. This question may be phrased in the language of extreme value
statistics and we establish some results towards a complete extreme value
law in this setting. The methods used are based on classical ideas from ge
ometry of numbers. This is work in progress\, joint with Keivan Mallahi-Ka
rai.\n
LOCATION:https://researchseminars.org/talk/NEDNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Varju (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20210322T161500Z
DTEND;VALUE=DATE-TIME:20210322T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/18
DESCRIPTION:Title: O
n the dimension of self-similar measures\nby Peter Varju (University o
f Cambridge) as part of New England Dynamics and Number Theory Seminar\n\n
Lecture held in Online.\n\nAbstract\nLet $f_1$\,…\,$f_n$ be a collection
of contracting similarities on $\\mathbb{R}$\, and let $p_1$\,…\,$p_n$
be a probability vector. There is a unique probability measure mu on $\\ma
thbb{R}$ that satisfies the identity\n$\\mu = p_1 f_1(\\mu) + … + p_n f_
n(\\mu)$.\nThis measure is called self-similar. The maps $f_1$\,…\,$f_n$
are said to satisfy the no exact overlaps condition if they generate a fr
ee semigroup (i.e. all compositions are distinct). Under this condition\,
the dimension of mu is conjectured to be the minimum of 1 and the ratio of
the entropy of $p_1$\,…\,$p_n$ and the average logarithmic contraction
factor of the $f_i$. This conjecture has been recently established in some
special cases\, including when $n=2$ and $f_1$ and $f_2$ have the same co
ntraction factor. In the talk I will discuss recent progress by Ariel Rapa
port and myself in the case $n=3$. In this case new difficulties arise as
was demonstrated by recent examples of Baker and Barany\, Kaenmaki of IFS
’s with arbitrarily weak separation properties.\n
LOCATION:https://researchseminars.org/talk/NEDNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tushar Das (University of Wisconsin - La Crosse)
DTSTART;VALUE=DATE-TIME:20210419T161500Z
DTEND;VALUE=DATE-TIME:20210419T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/19
DESCRIPTION:Title: U
sing templates to study problems in dynamics and number theory\nby Tus
har Das (University of Wisconsin - La Crosse) as part of New England Dynam
ics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nTemp
lates may be viewed as a combinatorial device that helps study\nasymptotic
properties of lattice successive minima. This simple idea\,\nintroduced i
n joint work with Lior Fishman\, David Simmons\, and Mariusz\nUrbanski\, p
romises to be useful in several areas beyond our current\napplications. Th
e latter lie at the fertile interface along Dani’s\ncorrespondence princ
iple between Diophantine approximation and\nhomogeneous flows\, deepened b
y Kleinbock & Margulis\; and Schmidt &\nSummerer’s parametric geometry o
f numbers\, deepened by Roy. Templates\nare at the heart of our variationa
l principle (arXiv:1901.06602)\,\nwhich provides a unified framework to co
mpute the Hausdorff and\npacking dimensions of a variety of sets of dynami
cal and\nnumber-theoretic interest. We will introduce and give some flavor
for\nour project\, hint at a few new directions\, and hope to present sev
eral\nopen problems of varying depth to reward participants of this\nwonde
rful seminar!\n
LOCATION:https://researchseminars.org/talk/NEDNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minju Lee (Yale University)
DTSTART;VALUE=DATE-TIME:20210308T171500Z
DTEND;VALUE=DATE-TIME:20210308T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/20
DESCRIPTION:Title: O
rbit closures of unipotent flows for hyperbolic manifolds with Fuchsian en
ds\nby Minju Lee (Yale University) as part of New England Dynamics and
Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nThis is joi
nt work with Hee Oh. We establish an analogue of Ratner’s orbit closure
theorem for any connected closed subgroup generated by unipotent elements
in $\\mathrm{SO}(d\,1)$ acting on the space $\\Gamma\\backslash\\mathrm{SO
}(d\,1)$\, assuming that the associated hyperbolic manifold $M=\\Gamma\\ba
ckslash\\mathbb{H}^d$ is a convex cocompact manifold with Fuchsian ends. F
or $d = 3$\, this was proved earlier by McMullen\, Mohammadi and Oh. In a
higher dimensional case\, the possibility of accumulation on closed orbits
of intermediate subgroups causes serious issues\, but in the end\, all or
bit closures of unipotent flows are relatively homogeneous. Our results im
ply the following: for any $k\\geq 1$\,\n(1) the closure of any $k$-horosp
here in $M$ is a properly immersed submanifold\;\n(2) the closure of any g
eodesic $(k+1)$-plane in $M$ is a properly immersed submanifold\;\n(3) an
infinite sequence of maximal properly immersed geodesic $(k+1)$-planes int
ersecting $\\mathrm{core} M$ becomes dense in $M$.\n
LOCATION:https://researchseminars.org/talk/NEDNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Han Yu (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20210412T161500Z
DTEND;VALUE=DATE-TIME:20210412T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/21
DESCRIPTION:Title: R
ational numbers near self-similar sets\nby Han Yu (University of Cambr
idge) as part of New England Dynamics and Number Theory Seminar\n\nLecture
held in Online.\n\nAbstract\nWe will discuss a problem on counting ration
al numbers near\nself-similar sets. In particular\, we will show that the
set of rational\nnumbers is ‘reasonably well distributed’ around the m
iddle $p$-th Cantor\nset when $p$ is a large integer. Our approach is via
Fourier analysis\nand we will also discuss some problems on Fourier transf
orm of\nself-similar measures which are of independent interest. As a resu
lt\, it\nis possible to show that $p=5$ satisfies the previous statement.
The\nmaterials come from various working-in-progress projects with D. Alle
n\,\nS. Chow and P. Varju.\n
LOCATION:https://researchseminars.org/talk/NEDNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Chevallier (Université de Haute Alsace)
DTSTART;VALUE=DATE-TIME:20210405T161500Z
DTEND;VALUE=DATE-TIME:20210405T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/22
DESCRIPTION:Title: M
inimal vectors in $\\C^2$ and best constant for Dirichlet theorem over $\\
C$\nby Nicolas Chevallier (Université de Haute Alsace) as part of New
England Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\n
Abstract\nWe study minimal vectors in lattices over Gaussian integers in $
\\C^2$.We show that the index of the sub-lattice generated by two consecut
ive minimal vectors in a lattice of $\\C^2$\, can be either $1$ or $2$.Nex
t\, we describe the constraints on pairs of consecutive minimal vectors. T
hese constraints make it possible to find the best constant for Dirichlet
theorem about approximations of complex numbers by quotient of Gaussian i
ntegers.\n
LOCATION:https://researchseminars.org/talk/NEDNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asaf Katz (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210426T161500Z
DTEND;VALUE=DATE-TIME:20210426T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/23
DESCRIPTION:Title: A
n application of Margulis’ inequality to effective equidistribution\
nby Asaf Katz (University of Michigan) as part of New England Dynamics and
Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nRatner’s
celebrated equidistribution theorem states that the trajectory of any poin
t in a homogeneous space under a unipotent flow is getting equidistributed
with respect to some algebraic measure. In the case where the action is h
orospherical\, one can deduce an effective equidistribution result by mixi
ng methods\, an idea that goes back to Margulis’ thesis. When the homoge
neous space is non-compact\, one needs to impose further “diophantine co
nditions” over the base point\, quantifying some recurrence rates\, in o
rder to get a quantified equidistribution result. In the talk I will discu
ss certain diophantine conditions\, and in particular I will show how a ne
w Margulis’ type inequality for translates of horospherical orbits helps
verify such conditions. This results in a quantified equidistribution res
ult for a large class of points\, akin to the results of A. Strombreggson
dealing with the \\textrm{SL}_2 case. In particular we deduce a fully effe
ctive quantitative equidistribution for horospherical trajectories of latt
ices defined over number fields\, without pertaining to the strong subspac
e theorem.\n
LOCATION:https://researchseminars.org/talk/NEDNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk
DTSTART;VALUE=DATE-TIME:20210329T161500Z
DTEND;VALUE=DATE-TIME:20210329T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/24
DESCRIPTION:by No Talk as part of New England Dynamics and Number Theory S
eminar\n\nLecture held in Online.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NEDNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pratyush Sarkar (Yale University)
DTSTART;VALUE=DATE-TIME:20210503T161500Z
DTEND;VALUE=DATE-TIME:20210503T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/25
DESCRIPTION:Title: G
eneralization of Selberg’s 3⁄16 theorem for convex cocompact thin subg
roups of SO(n\, 1)\nby Pratyush Sarkar (Yale University) as part of Ne
w England Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\
nAbstract\nSelberg’s 3/16 theorem for congruence covers of the modular s
urface is a beautiful theorem which has a natural dynamical interpretation
as uniform exponential mixing. Bourgain-Gamburd-Sarnak’s breakthrough w
orks initiated many recent developments to generalize Selberg’s theorem
for infinite volume hyperbolic manifolds. One such result is by Oh-Winter
establishing uniform exponential mixing for convex cocompact hyperbolic su
rfaces. These are not only interesting in and of itself but can also be us
ed for a wide range of applications including uniform resonance free regio
ns for the resolvent of the Laplacian\, affine sieve\, and prime geodesic
theorems. I will present a further generalization to higher dimensions and
some of these immediate consequences.\n
LOCATION:https://researchseminars.org/talk/NEDNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seungki Kim (University of Cincinnati)
DTSTART;VALUE=DATE-TIME:20210510T161500Z
DTEND;VALUE=DATE-TIME:20210510T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/26
DESCRIPTION:Title: C
ounting problems on a random lattice\nby Seungki Kim (University of Ci
ncinnati) as part of New England Dynamics and Number Theory Seminar\n\nLec
ture held in Online.\n\nAbstract\nA random lattice is a random element of
SL(n\,Z) \\ SL(n\,R) equipped with the probability measure inherited from
the Haar measure of SL(n\,R). Analogous to the usual lattice point-countin
g\, one tries to “count” — more precisely\, study the statistics of
— the random lattice points inside a ball or other shapes. I’ll give a
gentle introduction to this topic\, discussing the early works of Siegel\
, Rogers and Schmidt and some of the recent results\, as well as their app
lications.\n
LOCATION:https://researchseminars.org/talk/NEDNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayadev Athreya (University of Washington)
DTSTART;VALUE=DATE-TIME:20210923T161500Z
DTEND;VALUE=DATE-TIME:20210923T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/28
DESCRIPTION:Title: G
eometric Structures and Point Processes\nby Jayadev Athreya (Universit
y of Washington) as part of New England Dynamics and Number Theory Seminar
\n\nLecture held in Online.\n\nAbstract\nIn this talk\, we will prove the
convergence part of Khitchine’s theorem on non-degenerate manifolds. Thi
s confirms a conjecture of Kleinbock and Margulis in 1998. Our approach us
es geometric and dynamical ideas together with a new technique of `major a
nd minor arcs’. In particular\, we establish sharp upper bounds for the
number of rational points of bounded height lying near `major arcs’ and
give explicit exponentially small bounds for the measure of `minor arcs’
. This is joint work with Victor Beresnevich.\n
LOCATION:https://researchseminars.org/talk/NEDNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Hurtado (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210930T161500Z
DTEND;VALUE=DATE-TIME:20210930T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/29
DESCRIPTION:Title: H
eight Gap\, an Arithmetic Margulis Lemma and Almost Laws\nby Sebastian
Hurtado (University of Chicago) as part of New England Dynamics and Numbe
r Theory Seminar\n\nLecture held in Online.\n\nAbstract\nWe provide a new
(more elementary) proof of a result of E. Breuillard\, which state that a
set of matrices with algebraic entries generating a non-virtually solvable
group has a positive lower bound in its arithmetic height (we will explai
n this notion)\, this is a non-abelian version of Lehmer’s problem. We a
lso show that in arithmetic locally symmetric spaces\, short geodesics ten
d to be far from each other if the degree of the trace field is large. Thi
s lemma allows us to prove new results about growth of cohomology of seque
nces of locally symmetric spaces and to give a proof of a conjecture of Ge
lander. These results are works in progress with Joe Chen and Homin Lee\,
and with Mikolaj Fraczyk and Jean Raimbault.\n
LOCATION:https://researchseminars.org/talk/NEDNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dubi Kelmer (Boston College)
DTSTART;VALUE=DATE-TIME:20211021T161500Z
DTEND;VALUE=DATE-TIME:20211021T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/30
DESCRIPTION:Title: T
he light cone Siegel transform\, its moment formulas\, and their applicati
ons\nby Dubi Kelmer (Boston College) as part of New England Dynamics a
nd Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nIn this t
alk I will describe an analogue of the Siegel transform where the role of
Euclidean space is replaced by a light cone corresponding to an indefinite
quadratic form.In this case one can use results on the spectral theory of
incomplete Eisenstein series to establish moment formulas analogous to th
e classical formulas of Siegel\, Rogers\, and Schmidt.I will then describe
several applications of these formulas to counting lattice points on the
light cone\, as well as for the distribution of rational points on the sph
ere. All new results are based on joint work with Shucheng Yu.\n
LOCATION:https://researchseminars.org/talk/NEDNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lei Yang (Sichuan University)
DTSTART;VALUE=DATE-TIME:20211007T161500Z
DTEND;VALUE=DATE-TIME:20211007T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/31
DESCRIPTION:Title: K
hintchine’s theorem on manifolds\nby Lei Yang (Sichuan University) a
s part of New England Dynamics and Number Theory Seminar\n\nLecture held i
n Online.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NEDNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengyu Yang (ETH)
DTSTART;VALUE=DATE-TIME:20211014T161500Z
DTEND;VALUE=DATE-TIME:20211014T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/32
DESCRIPTION:Title: E
quidistribution of degenerate curves and Dirichlet improvability\nby P
engyu Yang (ETH) as part of New England Dynamics and Number Theory Seminar
\n\nLecture held in Online.\n\nAbstract\nIn the space of 3-lattices\, we s
tudy the translates of a line segment under a diagonal flow. Sharp conditi
ons for non-divergence and equidistribution will be given. As an applicati
on\, we will show that Lebesgue-almost every point on a planar line is Dir
ichlet non-improvable if and only if the line is irrational. This is joint
work with Kleinbock\, de Saxcé and Shah. Generalizations to higher dimen
sions will also be discussed (work in progress with Shah).\n
LOCATION:https://researchseminars.org/talk/NEDNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Demi Allen (University of Warwick)
DTSTART;VALUE=DATE-TIME:20211028T161500Z
DTEND;VALUE=DATE-TIME:20211028T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/33
DESCRIPTION:Title: A
n inhomogeneous Khintchine-Groshev Theorem without monotonicity\nby De
mi Allen (University of Warwick) as part of New England Dynamics and Numbe
r Theory Seminar\n\nLecture held in Online.\n\nAbstract\nThe classical (in
homogeneous) Khintchine-Groshev Theorem tells us that for a monotonic appr
oximating function $\\psi: \\mathbb{N} \\to [0\,\\infty)$ the Lebesgue mea
sure of the set of (inhomogeneously) $\\psi$-well-approximable points in $
\\mathbb{R}^{nm}$ is zero or full depending on\, respectively\, the conver
gence or divergence of $\\sum_{q=1}^{\\infty}{q^{n-1}\\psi(q)^m}$. In the
homogeneous case\, it is now known that the monotonicity condition on $\\p
si$ can be removed whenever $nm>1$ and cannot be removed when $nm=1$. In t
his talk I will discuss recent work with Felipe A. Ramírez (Wesleyan\, US
) in which we show that the inhomogeneous Khintchine-Groshev Theorem is tr
ue without the monotonicity assumption on $\\psi$ whenever $nm>2$. This re
sult brings the inhomogeneous theory almost in line with the completed hom
ogeneous theory. I will survey previous results towards removing monotonic
ity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem befo
re discussing the main ideas behind the proof our recent result.\n
LOCATION:https://researchseminars.org/talk/NEDNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Gorodnik (University of Zurich)
DTSTART;VALUE=DATE-TIME:20211104T161500Z
DTEND;VALUE=DATE-TIME:20211104T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/34
DESCRIPTION:Title: Q
uantitative equidistribution and Randomness\nby Alexander Gorodnik (Un
iversity of Zurich) as part of New England Dynamics and Number Theory Semi
nar\n\nLecture held in Online.\n\nAbstract\nWe discuss some results on qua
ntitative equidistribution on homogeneous spaces and related problems abou
t behaviour of arithmetic counting functions. This is a joint work with Bj
örklund and Fregoli.\n
LOCATION:https://researchseminars.org/talk/NEDNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Lutsko (Rutgers University)
DTSTART;VALUE=DATE-TIME:20211111T171500Z
DTEND;VALUE=DATE-TIME:20211111T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/35
DESCRIPTION:Title: P
air correlation of monomial sequences modulo 1\nby Chris Lutsko (Rutge
rs University) as part of New England Dynamics and Number Theory Seminar\n
\nLecture held in Online.\n\nAbstract\nFix $\\alpha\, \\theta > 0$\, and c
onsider the sequence $(\\alpha n^\\theta \\mod 1)_{n>0}$. Since the semina
l work of Rudnick-Sarnak (1998)\, and due to the Berry-Tabor conjecture in
quantum chaos\, the fine-scale properties of these dilated mononomial seq
uences have been intensively studied. In this talk\, I will briefly survey
what is known about these sequences and present a recent result (joint wi
th Sourmelidis and Technau) showing that for $\\theta \\le 1/3$\, and $\\a
lpha > 0$\, the pair correlation function is Poissonian. While the techniq
ues we use are derived from analytic number theory\, the problem is rooted
in dynamics and relates to dynamical proofs for related problems.\n
LOCATION:https://researchseminars.org/talk/NEDNT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Khayutin (Northwestern University)
DTSTART;VALUE=DATE-TIME:20211202T171500Z
DTEND;VALUE=DATE-TIME:20211202T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/36
DESCRIPTION:Title: T
wo-step equidistribution for bi-quadratic torus packets\nby Ilya Khayu
tin (Northwestern University) as part of New England Dynamics and Number T
heory Seminar\n\nLecture held in Online.\n\nAbstract\nA major challenge to
the asymptotic analysis of a sequence of probability measures on a homoge
neous space\, invariant under diagonalizable groups\, is the possibility o
f accumulation on intermediate homogeneous subspaces. In this aspect highe
r rank homogeneous flows cannot be expected to share the rigidity properti
es of unipotent ones. In particular\, the linearization technique fails fo
r diagonalizable flows. \n\nIn a joint work in progress with A. Wieser we
show how in favorable situations one can actually use the existence of int
ermediate homogeneous spaces in our benefit. We show that periodic measure
s on some packets of periodic torus orbits on PGL4(Z)\\PGL4(R) converge in
the limit to a measure with a non-trivial Haar component. The proof goes
by establishing high entropy for the limit measure. The method utilizes th
e intermediate homogeneous space to split the analysis into two more tract
able steps.\n
LOCATION:https://researchseminars.org/talk/NEDNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bersudsky (Technion)
DTSTART;VALUE=DATE-TIME:20211209T171500Z
DTEND;VALUE=DATE-TIME:20211209T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/37
DESCRIPTION:Title: O
n the image in the torus of sparse points on expanding analytic curves
\nby Michael Bersudsky (Technion) as part of New England Dynamics and Numb
er Theory Seminar\n\nLecture held in Online.\n\nAbstract\nIt is known that
the projection to the 2-torus of the normalised parameter measure on a ci
rcle of radius $R$ in the plane becomes uniformly distributed as $R$ grows
to infinity. I will discuss the following natural discrete analogue for t
his problem. Starting from an angle and a sequence of radii {$R_n$} which
diverges to infinity\, I will consider the projection to the 2-torus of th
e n’th roots of unity rotated by this angle and dilated by a factor of $
R_n$. The interesting regime in this problem is when $R_n$ is much larger
than n so that the dilated roots of unity appear sparsely on the dilated c
ircle.I will discuss 3 types of results:\n\nValidity of equidistribution f
or all angles when the sparsity is polynomial.\nFailure of equidistributio
n for some super polynomial dilations.\nEquidistribution for almost all an
gles for arbitrary dilations.\nI will discuss the above type of results in
greater generality and I will try to explain how the theory of o-minimal
structures is related to the proof.\n
LOCATION:https://researchseminars.org/talk/NEDNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Das (University of Houston)
DTSTART;VALUE=DATE-TIME:20220203T171500Z
DTEND;VALUE=DATE-TIME:20220203T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/38
DESCRIPTION:Title: A
n adelic version of the three gap theorem\nby Akshat Das (University o
f Houston) as part of New England Dynamics and Number Theory Seminar\n\nLe
cture held in Online.\n\nAbstract\nIn order to understand problems in dyna
mics which are sensitive to arithmetic properties of return times to regio
ns\, it is desirable to generalize classical results about rotations on th
e circle to the setting of rotations on adelic tori. One such result is th
e classical three gap theorem\, which is also referred to as the three dis
tance theorem and as the Steinhaus problem. It states that\, for any real
number\, a\, and positive integer\, N\, the collection of points na mod 1\
, where n runs from 1 to N\, partitions the circle into component arcs ha
ving one of at most three distinct lengths. Since the 1950s\, when this th
eorem was first proved independently by multiple authors\, it has been rep
roved numerous times and generalized in many ways. One of the more recent
proofs has been given by Marklof and Strömbergsson using a lattice based
approach to gaps problems in Diophantine approximation. In this talk\, we
use an adaptation of this approach to the adeles to prove a natural genera
lization of the classical three gap theorem for rotations on adelic tori.
This is joint work with Alan Haynes.\n
LOCATION:https://researchseminars.org/talk/NEDNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiyoung Han (TIFR)
DTSTART;VALUE=DATE-TIME:20220210T171500Z
DTEND;VALUE=DATE-TIME:20220210T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/39
DESCRIPTION:Title: T
he asymptotic distribution of the joint values of the integral lattice poi
nts for a system of a quadratic form and a linear form\nby Jiyoung Han
(TIFR) as part of New England Dynamics and Number Theory Seminar\n\nLectu
re held in Online.\n\nAbstract\nLet Q be a quadratic form and let L be a l
inear form on the n-dimensional real vector space. We are interested in th
e distribution of the image of the integral lattice under the map (Q\, L).
Developing the celebrated work of Eskin\, Margulis\, and Mozes in 1998\,
we provide the conditions of systems of forms which satisfy that the numbe
r of integral vectors in the ball of radius T whose joint values are conta
ined in a given bounded set converges asymptotically to the volume of the
region given by the level sets of the quadratic form and the linear form\,
intersecting with the ball of radius T\, as T goes to infinity. This cond
ition is introduced by Gorodnik in 2004.\nFor this\, we need to classify a
ll intermediate subgroups between the special orthogonal group preserving
Q and L and the special linear group. Among them\, only two closed subgrou
ps are of our concern. We will introduce Siegel integral formulas and equi
distribution theorems for each subgroup\, and show how to reach our main t
heorem. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.\n
LOCATION:https://researchseminars.org/talk/NEDNT/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julien Trevisan (Institut de Mathématiques de Jussieu)
DTSTART;VALUE=DATE-TIME:20220217T171500Z
DTEND;VALUE=DATE-TIME:20220217T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/40
DESCRIPTION:Title: L
imit laws in the lattice counting problem. The case of ellipses.\nby J
ulien Trevisan (Institut de Mathématiques de Jussieu) as part of New Engl
and Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstr
act\nLet E be an ellipse centered around 0. We are interested in the asymp
totic distribution\nof the error of the number of unimodular lattice point
s that fall into tE when the lattice is random\nand when t goes to infinit
y.\nBuilding on previous works by Bleher and by Fayad and Dolgopyat\, we s
how that the error term\, when normalized by the square root of t\, conver
ges in distribution towards an explicit distribution.\nFor this\, we first
use harmonic analysis to reduce the study of the normalized error to the
study of a Siegel transform that depends on t.\nThen\, and this is the key
part of our proof\, we show that\, when t goes to infinity\, this last Si
egel transform behaves in distribution as\, what we call\, a modified Sieg
el transform with random weights. Such objects often appear in average cou
nting problems.\nFinally\, we show that this last quantity converges almos
t surely\, and we study the existence of the moments of its law.\nThis wor
k was supervised by Bassam Fayad.\n
LOCATION:https://researchseminars.org/talk/NEDNT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irving Calderón (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20220303T171500Z
DTEND;VALUE=DATE-TIME:20220303T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/41
DESCRIPTION:Title: S
-adic quadratic forms and Homogeneous Dynamics\nby Irving Calderón (U
niversité Paris-Saclay) as part of New England Dynamics and Number Theory
Seminar\n\nLecture held in Online.\n\nAbstract\nWe present two new quanti
tative results about quadratic forms.\nLet $S = {\\infty} \\cup S_f$ be a
finite set of places of Q. Consider the ring $Z_S$ of S-integers\, and $Q_
S = \\prod{p \\in S} Q_p$. The first is a solution to the problem of decid
ing if any given integral quadratic forms $Q_1$ and $Q_2$ are $Z_S$-equiva
lent. The proof is based on a reformulation of the problem in terms of the
action of $O(Q_1\, Q_S)$ on the space $X{d\,S}$ of lattices of $Q_{S\,d}$
. A key tool are explicit mixing rates for the action of O(Q1\, QS) on clo
sed orbits in X{d\,S}. As an application we obtain\, for any S-integral or
thogonal group\, polynomial bounds on the S-norms of the elements of a fin
ite generating set.\nThese two results and the methods of proof are based
on the work of H. Li and G. Margulis for $S = { \\infty }$.\n
LOCATION:https://researchseminars.org/talk/NEDNT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nattalie Tamam (University of Michigan)
DTSTART;VALUE=DATE-TIME:20220310T171500Z
DTEND;VALUE=DATE-TIME:20220310T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/42
DESCRIPTION:Title: C
lassification of divergence of trajectories\nby Nattalie Tamam (Univer
sity of Michigan) as part of New England Dynamics and Number Theory Semina
r\n\nLecture held in Online.\n\nAbstract\nAs shown by Dani\, diophantine a
pproximations are in direct correspondence to the behavior of orbits in ce
rtain homogeneous spaces. We will discuss the interpretation of the diverg
ent trajectories and the obvious ones\, the ones diverging due to a purely
algebraic reason. As conjectured by Barak Weiss\, there is a complete cla
ssification of divergent trajectories when considering the action of subgr
oups of the diagonal group. We will discuss the last part of this conjectu
re\, showing that for a ‘large enough’ such subgroup\, every divergent
trajectory diverges obviously. This is a joint work with Omri Solan.\n
LOCATION:https://researchseminars.org/talk/NEDNT/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nate Hughes (University of Exeter)
DTSTART;VALUE=DATE-TIME:20220317T161500Z
DTEND;VALUE=DATE-TIME:20220317T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/43
DESCRIPTION:Title: E
ffective Counting and Spiralling of Lattice Approximates\nby Nate Hugh
es (University of Exeter) as part of New England Dynamics and Number Theor
y Seminar\n\nLecture held in Online.\n\nAbstract\nWe will prove an effecti
ve version of Dirichlet’s approximation theorem\, giving the error betwe
en the number of rational approximations to a real vector with denominator
less than some real number T and the asymptotic growth of this count. Add
itional results for linear forms can be obtained\, as well as results meas
uring the direction of these approximates\, known as ‘spiralling of latt
ice approximates’. These results are obtained by reformulating the numbe
r-theoretic problem to the context of homogeneous spaces of unimodular lat
tices. The advantage of this reformulation is that we have more tools to d
eal with the problem\, such as Siegel’s mean value theorem and Rogers’
higher moment formula. The proof involves using the ergodic properties of
diagonal flows on this homogeneous space to calculate the number of latti
ce approximates\, bounding the second moment of the count\, then applying
an effective ergodic theorem due to Gaposhkin. Particular attention is pai
d to the case of primitive lattices in two-dimensions\, where Rogers’ th
eorem fails. In this case\, we apply a new theorem by Kleinbock and Yu to
obtain a better error term than previous results due to Schmidt.\n
LOCATION:https://researchseminars.org/talk/NEDNT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Machado (the University of Cambridge)
DTSTART;VALUE=DATE-TIME:20220324T161500Z
DTEND;VALUE=DATE-TIME:20220324T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/44
DESCRIPTION:Title: S
uperrigidity and arithmeticity for some aperiodic subsets in higher-rank s
imple Lie groups\nby Simon Machado (the University of Cambridge) as pa
rt of New England Dynamics and Number Theory Seminar\n\nLecture held in On
line.\n\nAbstract\nMeyer sets are fascinating objects: they are aperiodic
subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic
order. Sets of vertices of the Penrose tiling (P3) and Pisot-Vijarayaghava
n numbers of a real number field are some of the most well-known examples.
In his pioneering work\, Meyer provided a powerful and elegant characteri
sation of Meyer sets. Years later\, Lagarias proved a similar characterisa
tion starting from what seemed to be considerably weaker assumptions.\nA f
ascinating question asks whether Meyer’s and Lagarias’ results may be
extended to more general ambient groups. In fact\, a first result in that
direction was already obtained in Meyer’s work: he proved a sum-product
phenomenon which\, implicitly\, boiled down to a classification of Meyer s
ets in the group of affine transformations of the line.\nI will talk about
a generalisation of both Meyer’s and Lagarias’ theorems to discrete s
ubsets of higher-rank simple Lie groups. I will explain how this result ca
n be seen as a generalisation of Margulis’ arithmeticity theorem and how
it can be deduced from Zimmer’s cocycle superrigidity. We will see that
\, surprisingly\, Pisot-Vijarayaghavan numbers appear naturally in this co
ntext too.\n
LOCATION:https://researchseminars.org/talk/NEDNT/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johannes Schleischitz (Middle East Technical University)
DTSTART;VALUE=DATE-TIME:20220331T161500Z
DTEND;VALUE=DATE-TIME:20220331T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/45
DESCRIPTION:Title: E
xact uniform approximation and Dirichlet spectrum\nby Johannes Schleis
chitz (Middle East Technical University) as part of New England Dynamics a
nd Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nWe consid
er the Dirichlet spectrum\, with respect to maximum norm and simultaneous
approximation. It is basically the analogue of the famous (multi-dimension
al) Lagrange spectrum with respect to uniform approximation. By Dirichlet
’s Theorem it is contained in [0\,1]. The central new result is that it
equals the entire interval [0\,1] when the number of variables is two or m
ore. We thereby get a new\, constructive proof of a recent result by Beres
nevich\, Guan\, Marnat\, Ramirez and Velani that there are Dirichlet impro
vable vectors that are neither bad nor singular\, in any dimension. We pro
vide several generalizations\, including metrical claims.\n
LOCATION:https://researchseminars.org/talk/NEDNT/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilio Corso (ETH)
DTSTART;VALUE=DATE-TIME:20220407T161500Z
DTEND;VALUE=DATE-TIME:20220407T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/46
DESCRIPTION:Title: A
symptotics of the equidistribution rate of expanding circles on compact hy
perbolic quotients and applications\nby Emilio Corso (ETH) as part of
New England Dynamics and Number Theory Seminar\n\nLecture held in Online.\
n\nAbstract\nEquidistribution properties of translates of orbits for subgr
oup actions on homogeneous spaces are intimately linked to the mixing feat
ures of the global action of the ambient group. The connection appears alr
eady in Margulis’ thesis (1969)\, displaying its full potential in the w
ork of Eskin and McMullen during the nineties. On a quantitative level\, t
he philosophy underpinning this linkage allows to transfer mixing rates to
effective estimates for the rate of equidistribution\, albeit at the cost
of a sizeable loss in the exponent. In joint work with Ravotti\, we inste
ad resort to a spectral method\, pioneered by Ratner in her study of quant
itative mixing of geodesic and horocycle flows\, in order to obtain the pr
ecise asymptotic behaviour of averages of regular observables along expand
ing circles on compact hyperbolic surfaces. The primary goal of the talk i
s to outline the salient traits of this method\, illustrating how it leads
to the relevant asymptotic expansion. In addition\, we shall also present
applications of the main result to distributional limit theorems and to q
uantitative error estimates on the corresponding hyperbolic lattice point
counting problem\, the latter having been examined\, to date\, only throug
h number-theoretical methods in works of Selberg\, Lax-Phillips and Philli
ps-Rudnick.\n
LOCATION:https://researchseminars.org/talk/NEDNT/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikolaj Fraczyk (University of Chicago)
DTSTART;VALUE=DATE-TIME:20220414T161500Z
DTEND;VALUE=DATE-TIME:20220414T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/47
DESCRIPTION:Title: T
hin part of the arithmetic orbifolds\nby Mikolaj Fraczyk (University o
f Chicago) as part of New England Dynamics and Number Theory Seminar\n\nLe
cture held in Online.\n\nAbstract\nLet X be a symmetric space. The collar
lemma\, also known as the Margulis lemma\, says that there exists an epsil
on=epsilon(X)\, such that the epsilon-thin part of a locally symmetric spa
ce X/\\Gamma looks locally like a quotient by a virtually unipotent subgro
up. It turns out that in the arithmetic setting we can improve this lemma
by making the epsilon grow linearly in the degree of the number filed gene
rated by the traces of elements of \\Gamma. I will explain why this is the
case and present several applications\, including the proof of the fact t
hat an arithmetic locally symmetric manifold M is homotopy equivalent to a
simplicial complex of size bounded linearly in the volume of M and degree
s of all vertices bounded uniformly in terms of X. Based on a joint work w
ith Sebastian Hurtado and Jean Raimbault.\n
LOCATION:https://researchseminars.org/talk/NEDNT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Hoover (Boston College)
DTSTART;VALUE=DATE-TIME:20220428T161500Z
DTEND;VALUE=DATE-TIME:20220428T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/48
DESCRIPTION:Title: E
ffective Equidistribution on Hilbert Modular Surfaces\nby Ian Hoover (
Boston College) as part of New England Dynamics and Number Theory Seminar\
n\nLecture held in Online.\n\nAbstract\nWhile ineffective equidistribution
has been understood much more generally\, effective results for non-compa
ct orbits have been more scarce. I will give effective (polynomial) error
rates for the translates of diagonal orbits on Hilbert modular surfaces. T
his work follows as a higher dimensional extension of the work of Kelmer a
nd Kontorovich.\n
LOCATION:https://researchseminars.org/talk/NEDNT/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiajie Zheng (Brandeis University)
DTSTART;VALUE=DATE-TIME:20220505T161500Z
DTEND;VALUE=DATE-TIME:20220505T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/49
DESCRIPTION:Title: D
ynamical Borel–Cantelli Lemma for Lipschitz Twists\nby Jiajie Zheng
(Brandeis University) as part of New England Dynamics and Number Theory Se
minar\n\nLecture held in Online.\n\nAbstract\nIn the study of some dynamic
al systems\, the limit superior of a sequence of measurable sets is often
of interest. The shrinking targets and recurrence are two of the most comm
only studied problems that concern limit superior sets. However\, the zero
-one laws for the shrinking targets and recurrence are usually treated sep
arately and proved differently. In this talk\, we construct a generalized
definition that can specialize into the shrinking targets and recurrence a
nd our approach gives a unified proof to the zero-one laws for the two pro
blems.\n
LOCATION:https://researchseminars.org/talk/NEDNT/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Baker (University of Birmingham)
DTSTART;VALUE=DATE-TIME:20220922T161500Z
DTEND;VALUE=DATE-TIME:20220922T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/50
DESCRIPTION:Title: O
verlapping iterated function systems from the perspective of Metric Number
Theory\nby Simon Baker (University of Birmingham) as part of New Engl
and Dynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstr
act\nKhintchine’s theorem is a classical result from metric number theor
y which relates the Lebesgue measure of certain limsup sets with the diver
gence of naturally occurring volume sums. Importantly this result provides
a quantitative description of how the rationals are distributed within th
e reals. In this talk I will discuss some recent work where I prove that a
similar Khintchine like phenomenon occurs typically within many families
of overlapping iterated function systems. Families of iterated function sy
stems these results apply to include those arising from Bernoulli convolut
ions\, the 0\,1\,3 problem\, and affine contractions with varying translat
ion parameters. \nTime permitting I also will discuss a particular family
of iterated function systems for which we can be more precise. Our analysi
s of this family shows that by studying the metric properties of limsup se
ts\, we can distinguish between the overlapping behaviour of iterated func
tion systems in a way that is not available to us by simply studying prope
rties of self-similar measures.\n
LOCATION:https://researchseminars.org/talk/NEDNT/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juno Seong (UC San Diego)
DTSTART;VALUE=DATE-TIME:20221006T161500Z
DTEND;VALUE=DATE-TIME:20221006T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/51
DESCRIPTION:Title: A
n avoidance principle and Margulis functions for expanding translates of u
nipotent orbits\nby Juno Seong (UC San Diego) as part of New England D
ynamics and Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\n
Avoidance principles — quantifying how much time trajectories avoid cert
ain subsets of the ambient space — have been fruitful in the study of dy
namical systems. We prove an avoidance principle for expanding translates
of unipotent orbits for some semisimple homogeneous spaces. In addition\,
we prove a quantitative isolation result of closed orbits and give an uppe
r bound on the number of closed orbits of bounded volume. The proof of our
results relies on the construction of a Margulis function and the theory
of finite dimensional representations of semisimple Lie groups. This is jo
int work with Anthony Sanchez.\n
LOCATION:https://researchseminars.org/talk/NEDNT/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Lutsko (Rutgers University)
DTSTART;VALUE=DATE-TIME:20221013T161500Z
DTEND;VALUE=DATE-TIME:20221013T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/52
DESCRIPTION:Title: A
Spectral Approach to Counting and Equidistribution\nby Chris Lutsko (
Rutgers University) as part of New England Dynamics and Number Theory Semi
nar\n\nLecture held in Online.\n\nAbstract\nSince the early 20th century\,
spectral methods have been used to obtain effective counting theorems for
various objects of interest in number theory\, geometry and group theory.
In this talk I’ll start by introducing two classical problems: the Gaus
s circle problem\, and the Apollonian counting problem. By surveying resul
ts on these problems (and some generalizations)\, I’ll demonstrate how t
o use spectral methods to obtain effective asymptotics for some very class
ical problems. Then I will try and explain how to generalize this method t
o apply to certain horospherical equidistribution theorems.\n
LOCATION:https://researchseminars.org/talk/NEDNT/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lam Pham (Brandeis University)
DTSTART;VALUE=DATE-TIME:20221020T161500Z
DTEND;VALUE=DATE-TIME:20221020T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/53
DESCRIPTION:Title: S
hort closed geodesics in higher rank arithmetic locally symmetric spaces\nby Lam Pham (Brandeis University) as part of New England Dynamics and
Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nA well-known
conjecture of Margulis predicts that there is a uniform lower bound on th
e systole of any irreducible arithmetic locally symmetric space. Recently\
, in joint work with Mikolaj Fraczyk\, we show that for simple Lie groups
of higher rank\, this conjecture is equivalent to a well-known conjecture
in number theory: that Salem numbers are uniformly bounded away from 1. I
will discuss our proof and some tools used\, and some additional results w
hich hold unconditionally and highlight the structure of the bottom of the
length spectrum.\n
LOCATION:https://researchseminars.org/talk/NEDNT/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreyasi Datta (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221103T161500Z
DTEND;VALUE=DATE-TIME:20221103T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/54
DESCRIPTION:Title: p
-Adic Diophantine approximation with respect to fractal measures\nby S
hreyasi Datta (University of Michigan) as part of New England Dynamics and
Number Theory Seminar\n\nLecture held in Online.\n\nAbstract\nI will give
an introduction to Diophantine approximation problems starting with the f
amous Sprindzuk Conjecture (which is now a theorem by Kleinbock and Margul
is\, who solved this using homogeneous dynamics).\nNext\, I will talk abou
t p-adic Diophantine approximation and how it is different than the real c
ase. In a very recent work with Anish Ghosh and Victor Beresnevich we solv
ed a conjecture of Kleinbock and Tomanov\, which shows pushforward of a fr
actal measure by ‘nice’ functions exhibits ‘nice’ Diophantine prop
erties. In particular\, we prove p-adic analogue of a result by Kleinbock\
, Lindenstrauss and Weiss on friendly measures. I will talk about how lack
of the mean value theorem makes life difficult in the p-adic fields. (No
prior knowledge on this subject will be assumed!)\n
LOCATION:https://researchseminars.org/talk/NEDNT/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Moshchevitin (Lomonosov Moscow State University)
DTSTART;VALUE=DATE-TIME:20221110T171500Z
DTEND;VALUE=DATE-TIME:20221110T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/55
DESCRIPTION:Title: O
n inhomogeneous Diophantine approximation\nby Nikolay Moshchevitin (Lo
monosov Moscow State University) as part of New England Dynamics and Numbe
r Theory Seminar\n\nLecture held in Online.\n\nAbstract\nWe will discuss s
ome classical and modern results related to systems of inhomogeneous linea
r forms. We will begin with Kronecker approximation theorem and famous re
sults by Khintchine and continue with rather modern problems\, in particul
ar related to weighted setting and coprime approximation.\n
LOCATION:https://researchseminars.org/talk/NEDNT/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas de Saxce (Université Paris-Nord)
DTSTART;VALUE=DATE-TIME:20221117T171500Z
DTEND;VALUE=DATE-TIME:20221117T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/56
DESCRIPTION:Title: R
ational approximations to linear subspaces\nby Nicolas de Saxce (Unive
rsité Paris-Nord) as part of New England Dynamics and Number Theory Semin
ar\n\nLecture held in Online.\n\nAbstract\nUsing diagonal orbits on the sp
ace of lattices\, we revisit some old questions of Schmidt concerning diop
hantine approximation on Grassmann varieties\, and in particular\, we prov
e a version of Dirichlet’s principle in that setting.\n
LOCATION:https://researchseminars.org/talk/NEDNT/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tariq Osman (Brandeis University)
DTSTART;VALUE=DATE-TIME:20221201T171500Z
DTEND;VALUE=DATE-TIME:20221201T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/57
DESCRIPTION:Title: T
ail Asymptotics for Generalised Theta Sums with Rational Parameters\nb
y Tariq Osman (Brandeis University) as part of New England Dynamics and Nu
mber Theory Seminar\n\nLecture held in Online.\n\nAbstract\nWe define gene
ralised theta sums as exponential sums of the form S^f_N(x\; \\alpha\, \\b
eta) := \\sum_{n \\in \\mathbb Z} f(n/N) e((1/2 n^2 + \\beta n)x + \\alpha
n)\, where e(z) = e^{2 \\pi i z}. If \\alpha and \\beta are fixed real nu
mbers\, and x is chosen randomly from the unit interval\, we may use homog
eneous dynamics to show that N^{-1/2} S^f_N$ possesses a limiting distribu
tion as N goes to infinity\, provided f is sufficiently regular. In joint
work with F. Cellarosi\, we prove that for specific rational pairs (\\alph
a\, \\beta) this limiting distribution is compactly supported and that all
other rational pairs lead to a limiting distribution with heavy tails. Th
is complements the existing work of F. Cellarosi and J. Marklof where at l
east one of \\alpha or \\beta is irrational.\n
LOCATION:https://researchseminars.org/talk/NEDNT/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART;VALUE=DATE-TIME:20221208T171500Z
DTEND;VALUE=DATE-TIME:20221208T183000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/58
DESCRIPTION:Title: C
ounting rationals and diophantine approximation on fractals\nby Sam Ch
ow (University of Warwick) as part of New England Dynamics and Number Theo
ry Seminar\n\nLecture held in Online.\n\nAbstract\nWe count rationals in m
issing-digit sets\, with applications to diophantine approximation. In the
process\, we develop the theory of Fourier \\ell^1 dimension\, including
the computational aspect.\n
LOCATION:https://researchseminars.org/talk/NEDNT/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Omri Solan (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20230921T161500Z
DTEND;VALUE=DATE-TIME:20230921T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/59
DESCRIPTION:Title: B
irkhoff generic points on curves\nby Omri Solan (Hebrew University of
Jerusalem) as part of New England Dynamics and Number Theory Seminar\n\nLe
cture held in Online.\n\nAbstract\nLet $a_t$ be a diagonal flow on the spa
ce X of unimodular lattices in R^n. A point x in X is called Birkhoff gene
ric if a_t.x equidistributes in X as t\\to \\infty. By Birkhoff ergodic th
eorem\, almost every point x in X is Birkhoff generic. One may ask whether
the same is true when the point x is sampled according to a measure singu
lar to Lebesgue. \nIn a joint work with Andreas Wieser\, we discuss the ca
se of a generic point x in an analytic curve in X\, and show that under ce
rtain conditions\, it must be Birkhoff generic. This Birkhoff genericity r
esult has various applications in Diophantine approximation. In this talk
we will relate Birkhoff genericity to approximations of real numbers by al
gebraic numbers of degree at most n.\n
LOCATION:https://researchseminars.org/talk/NEDNT/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zach Selk (Queen’s University)
DTSTART;VALUE=DATE-TIME:20230928T161500Z
DTEND;VALUE=DATE-TIME:20230928T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/60
DESCRIPTION:Title: S
tochastic Calculus for the Theta Process\nby Zach Selk (Queen’s Univ
ersity) as part of New England Dynamics and Number Theory Seminar\n\nLectu
re held in Online.\n\nAbstract\nThe Theta process\, $X(t)$\, is a complex
valued stochastic process of number theoretical origin arising as a scalin
g limit of quadratic Weyl sums $$\\sum_{n=1}^N e^{2\\pi i \\left(\\frac{1}
{2}(n^2+\\beta)x+\\alpha n\\right)}\,$$ where $(\\alpha\,\\beta)\\in \\mat
hbb R^2 \\setminus \\mathbb Q^2$ and $x\\in \\mathbb R$ is chosen at rando
m according to any law absolutely continuous with respect to Lebesgue meas
ure. The Theta process can be explicitly represented as $X(t)=\\sqrt{t} \\
Theta(\\Gamma g \\Phi^{2 \\log t})$ where $\\Theta$ is an automorphic func
tion defined on Lie group $G$\, invariant under left multiplication under
lattice $\\Gamma$. Additionally\, $g\\in \\Gamma \\setminus G$ is chosen H
aar uniformly at random and $\\Phi$ is the geodesic flow on $\\Gamma \\set
minus G$. The Theta process shares several similar properties with the Bro
wnian motion. In particular\, both lack differentiability and have the sam
e $p$ variation and H\\”older properties.\nSimilarly to Brownian motion\
, standard calculus and even Young/Riemann-Stieltjes calculus techniques d
o not work. However\, Brownian motion is what is known as a martingale all
owing for a classical theory of It\\^o calculus which makes use of cancell
ations “on average”. The It\\^o calculus can be used to prove several
properties of Brownian motion such as its conformal invariance\, bounds on
its running maximum in terms of its quadratic variation\, absolutely cont
inuous changes in measure and much more. \nUnfortunately\, we show that th
e Theta process $X$ is not a (semi)martingale\, therefore It\\^o technique
s don’t work. However\, a new theory introduced in 1998 by Terry Lyons c
alled rough paths theory handles processes with the same analytic regulari
ty as $X$. The key idea in rough paths theory is that constructing stochas
tic calculus for a signal can be reduced to constructing the “iterated i
ntegrals” of the signal. In this talk\, we will show the construction of
the iterated integrals – the “rough path” – above the process $X$
. Joint with Francesco Cellarosi.\n
LOCATION:https://researchseminars.org/talk/NEDNT/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuval Yifrach (Technion)
DTSTART;VALUE=DATE-TIME:20231005T161500Z
DTEND;VALUE=DATE-TIME:20231005T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232658Z
UID:NEDNT/61
DESCRIPTION:Title: A
variation on the p-adic Littlewood Conjecture\nby Yuval Yifrach (Tech
nion) as part of New England Dynamics and Number Theory Seminar\n\nLecture
held in Online.\n\nAbstract\nWe consider a variation on the p-adic Little
wood Conjecture where instead of using powers of one prime\, we use arbitr
arily large primes. We examine this conjecture from two viewpoints: the Di
ophantine-approximation one and the dynamical one. Using the dynamical vie
wpoint\, we rephrase the conjecture using Hecke neighbors and prove a part
ial statement towards the conjecture. Namely\, we prove that the Hausdorff
dimension of the exception set is strictly smaller than 1. Our tools for
the proof are mainly the effective equidistribution of Hecke neighbors due
to Oh et al and to expander properties of $SL_2(Z/pZ)$ due to Bourgain-Ga
mburd. This talk is based on an ongoing joint work with Erez Nesharim.\n
LOCATION:https://researchseminars.org/talk/NEDNT/61/
END:VEVENT
END:VCALENDAR