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BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART;VALUE=DATE-TIME:20220401T103000Z
DTEND;VALUE=DATE-TIME:20220401T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/1
DESCRIPTION:Title: Co
unting rationals and diophantine approximation on fractals\nby Sam Cho
w (University of Warwick) as part of Number theory by the sea\n\n\nAbstrac
t\nWe discuss the problem of counting rationals on fractals\, with\napplic
ations to diophantine approximation. In the process\, we develop the\ntheo
ry of the Fourier $\\ell^1$ dimension including\, for Bernoulli\nmeasures\
, its effective computation via induction on scales. Joint with\nDemi Alle
n\, P´eter Varj´u and Han Yu.\n
LOCATION:https://researchseminars.org/talk/ntsea/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel El-Baz (TU Graz)
DTSTART;VALUE=DATE-TIME:20220408T103000Z
DTEND;VALUE=DATE-TIME:20220408T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/8
DESCRIPTION:Title: Pr
imitive rational points on expanding horospheres: effective joint equidist
ribution\nby Daniel El-Baz (TU Graz) as part of Number theory by the s
ea\n\n\nAbstract\nI will report on ongoing work with Min Lee and Andreas\n
Strömbergsson. Using techniques from analytic number theory\, spectral\nt
heory\, geometry of numbers as well as a healthy dose of linear algebra\na
nd building on a previous work by Bingrong Huang\, Min Lee and myself\, we
\nfurnish a new proof of a 2016 theorem by Einsiedler\, Mozes\, Shah and\n
Shapira. That theorem concerns the equidistribution of primitive rational\
npoints on certain manifolds and our proof has the added benefit of\nyield
ing a rate of convergence. It turns out to have (perhaps surprising)\nappl
ications to the theory of random graphs\, which I shall also discuss.\n
LOCATION:https://researchseminars.org/talk/ntsea/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raphael Steiner (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20220422T103000Z
DTEND;VALUE=DATE-TIME:20220422T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/10
DESCRIPTION:Title: S
up-norms and fourth moment of automorphic forms\nby Raphael Steiner (E
TH Zurich) as part of Number theory by the sea\n\n\nAbstract\nThe study of
sup-norms of eigenfunctions of the Laplacian is a classical problem in ha
rmonic analysis. In an arithmetic setting\, they find further applications
to L-functions and geometric questions of the underlying spaces\, such as
Diophantine approximation and diameters. We discuss how they have been st
udied in the past and how a new approach allows one to study a higher (fou
rth) moment. In joint work with Ilya Khayutin and Paul Nelson\, we focus o
n the volume aspect and improve upon prior work by Templier\, Harcos-Templ
ier\, Blomer-Michel\, Toma.\n
LOCATION:https://researchseminars.org/talk/ntsea/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kummari Mallesham (ISI Kolkata)
DTSTART;VALUE=DATE-TIME:20220429T103000Z
DTEND;VALUE=DATE-TIME:20220429T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/11
DESCRIPTION:Title: G
eneral Rankin-Selberg problem\nby Kummari Mallesham (ISI Kolkata) as p
art of Number theory by the sea\n\n\nAbstract\nIn analytic number theory\,
it is a very fundamental question\nto understand the average order of an
arithmetic function $a(n)$. In this talk\, we discuss bounds for the avera
ge order when the $a(n)$'s\nare given by coefficients of Rankin-Selberg L-
functions of holomorphic cusp forms\n$f$ and $g$. The content of the talk
is based on ongoing work with Aritra\nGhosh\, Ritabrata Munshi and Saurabh
Kumar Singh.\n
LOCATION:https://researchseminars.org/talk/ntsea/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Subhajit Jana (Max Planck Institute for Mathematics Bonn)
DTSTART;VALUE=DATE-TIME:20220513T103000Z
DTEND;VALUE=DATE-TIME:20220513T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/13
DESCRIPTION:Title: A
lmost optimal Diophantine exponent for $\\mathrm{SL}(n)$\nby Subhajit
Jana (Max Planck Institute for Mathematics Bonn) as part of Number theory
by the sea\n\n\nAbstract\nWe will start by describing the density of $\\ma
thrm{SL}_n(\\mathbb{Z}[1/p])$ in $\\mathrm{SL}_n(\\mathbb{R})$ in a quanti
tative manner along the line of work by Ghosh--Gorodnik--Nevo. The Diophan
tine exponent $\\kappa$ for a pair of elements $x\,y \\in\n\\mathrm{SL}_n(
\\mathbb{R})$ is a certain positive real number that\, loosely\, measures
the complexity of an element $\\gamma\\in\\mathrm{SL}_n(\\mathbb{Z}[1/p])$
such that $\\gamma x$ approximates $y$ with a prescribed error. Ghosh--Go
rodnik--Nevo\nconjectured that $\\kappa$ should be optimal\, which means $
\\kappa \\le 1$ (after certain normalization)\, and proved this on certain
\nvarieties. However\, for $\\mathrm{SL}(n)$ their method gives $\\kappa \
\le n-1$. In this talk\, we try to describe how certain automorphic\ntechn
iques can improve the bound of $\\kappa$ to something as\ngood as $1+O(1/n
)$. If time permits\, we will also talk about the $L^2$-growth of the Eise
nstein series on reductive groups. This is one of the inputs in our proof
towards improved Diophantine exponent. This is a joint work with Amitay Ka
mber.\n
LOCATION:https://researchseminars.org/talk/ntsea/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (University of Göttingen)
DTSTART;VALUE=DATE-TIME:20220520T103000Z
DTEND;VALUE=DATE-TIME:20220520T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/14
DESCRIPTION:Title: D
ensity of rational points near/on compact manifolds with certain curvature
conditions\nby Damaris Schindler (University of Göttingen) as part o
f Number theory by the sea\n\n\nAbstract\nIn this talk I will discuss join
t work with Shuntaro Yamagishi where we\nestablish an asymptotic formula f
or the number of rational points\, with bounded\ndenominators\, within a g
iven distance to a compact submanifold M of R^n with a\ncertain curvature
condition. Technically we build on work of Huang on the density of\nration
al points near hypersurfaces. One of our goals is to explore generalisatio
ns\nto higher codimension. In particular we show that assuming certain cur
vature\nconditions in codimension at least two\, leads to upper bounds for
the number of\nrational points on M which are even stronger than what wou
ld be predicted by the\nanalogue of Serre's dimension growth conjecture.\n
LOCATION:https://researchseminars.org/talk/ntsea/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART;VALUE=DATE-TIME:20220603T103000Z
DTEND;VALUE=DATE-TIME:20220603T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/15
DESCRIPTION:Title: A
dditive equations over lattice points on spheres\nby Akshat Mudgal (Un
iversity of Oxford) as part of Number theory by the sea\n\n\nAbstract\nIn
this talk\, we will consider additive properties of lattice points on\nsph
eres. Thus\, defining $S_m$ to be the set of lattice points on the sphere
$x^2 + y^2\n+ z^2 + w^2 = m$\, we are interested in counting the number of
solutions to the\nequation $a_1 + a_2 = a_3 + a_4\,$ where $a_1\, ...\, a
_4$ lie in some arbitrary subset $A$ of $S_m$. Such an inquiry is closely
related to various problems in harmonic analysis and analytic number theor
y\, including Bourgain's discrete restriction conjecture for spheres. We w
ill survey some recent results in this direction\, as well as describe som
e of the various\ntechniques\, arising from areas such as incidence geomet
ry\, analytic number theory\nand arithmetic combinatorics\, that have been
employed to tackle this type of\nproblem.\n
LOCATION:https://researchseminars.org/talk/ntsea/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Ostafe (The University of New South Wales)
DTSTART;VALUE=DATE-TIME:20220610T103000Z
DTEND;VALUE=DATE-TIME:20220610T113000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073927Z
UID:ntsea/16
DESCRIPTION:Title: I
nteger matrices with a given characteristic polynomial and multiplicative
dependence of matrices\nby Alina Ostafe (The University of New South W
ales) as part of Number theory by the sea\n\n\nAbstract\nWe consider the s
et $\\mathcal{M}_n(\\mathbb{Z}\; H))$ of $n\\times n$-matrices with\ninteg
er elements of size at most $H$ and obtain upper and lower bounds on the n
umber\nof $s$-tuples\nof matrices from $\\mathcal{M}_n(\\mathbb{Z}\; H)$\,
satisfying various multiplicative\nrelations\, including\nmultiplicative
dependence\, commutativity and\nbounded generation of a subgroup of $\\mat
hrm{GL}_n(\\mathbb{Q})$. These problems\ngeneralise those studied\nin the
scalar case $n=1$ by F. Pappalardi\, M. Sha\, I. E. Shparlinski and C. L.\
nStewart (2018) with an\nobvious distinction due to the non-commutativity
of matrices.\nAs a part of our method\, we obtain a new upper bound on the
number of matrices from\n$\\mathcal{M}_n(\\mathbb{Z}\; H)$\nwith a given
characteristic polynomial $f \\in\\mathbb{Z}[X]$\, which is uniform with\n
respect to $f$. This complements\nthe asymptotic formula of A. Eskin\, S.
Mozes and N. Shah (1996) in which $f$ has to\nbe fixed and irreducible.\n\
nJoint work with Igor Shparlinski.\n
LOCATION:https://researchseminars.org/talk/ntsea/16/
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