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BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART:20220401T103000Z
DTEND:20220401T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /1/">Counting rationals and diophantine approximation on fractals</a>\nby 
 Sam Chow (University of Warwick) as part of Number theory by the sea\n\n\n
 Abstract\nWe discuss the problem of counting rationals on fractals\, with\
 napplications to diophantine approximation. In the process\, we develop th
 e\ntheory of the Fourier $\\ell^1$ dimension including\, for Bernoulli\nme
 asures\, its effective computation via induction on scales. Joint with\nDe
 mi Allen\, P´eter Varj´u and Han Yu.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel El-Baz (TU Graz)
DTSTART:20220408T103000Z
DTEND:20220408T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /8/">Primitive rational points on expanding horospheres: effective joint e
 quidistribution</a>\nby Daniel El-Baz (TU Graz) as part of Number theory b
 y the sea\n\n\nAbstract\nI will report on ongoing work with Min Lee and An
 dreas\nStrömbergsson. Using techniques from analytic number theory\, spec
 tral\ntheory\, geometry of numbers as well as a healthy dose of linear alg
 ebra\nand building on a previous work by Bingrong Huang\, Min Lee and myse
 lf\, we\nfurnish a new proof of a 2016 theorem by Einsiedler\, Mozes\, Sha
 h and\nShapira. That theorem concerns the equidistribution of primitive ra
 tional\npoints on certain manifolds and our proof has the added benefit of
 \nyielding a rate of convergence. It turns out to have (perhaps surprising
 )\napplications to the theory of random graphs\, which I shall also discus
 s.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raphael Steiner (ETH Zurich)
DTSTART:20220422T103000Z
DTEND:20220422T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /10/">Sup-norms and fourth moment of automorphic forms</a>\nby Raphael Ste
 iner (ETH Zurich) as part of Number theory by the sea\n\n\nAbstract\nThe s
 tudy of sup-norms of eigenfunctions of the Laplacian is a classical proble
 m in harmonic analysis. In an arithmetic setting\, they find further appli
 cations to L-functions and geometric questions of the underlying spaces\, 
 such as Diophantine approximation and diameters. We discuss how they have 
 been studied in the past and how a new approach allows one to study a high
 er (fourth) moment. In joint work with Ilya Khayutin and Paul Nelson\, we 
 focus on the volume aspect and improve upon prior work by Templier\, Harco
 s-Templier\, Blomer-Michel\, Toma.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kummari Mallesham (ISI Kolkata)
DTSTART:20220429T103000Z
DTEND:20220429T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /11/">General Rankin-Selberg problem</a>\nby Kummari Mallesham (ISI Kolkat
 a) as part of Number theory by the sea\n\n\nAbstract\nIn analytic number t
 heory\, it is a very fundamental question\nto understand the average order
  of an arithmetic function $a(n)$. In this talk\, we discuss bounds for th
 e average order when the $a(n)$'s\nare given by coefficients of Rankin-Sel
 berg L-functions of holomorphic cusp forms\n$f$ and $g$. The content of th
 e talk is based on ongoing work with Aritra\nGhosh\, Ritabrata Munshi and 
 Saurabh Kumar Singh.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Subhajit Jana (Max Planck Institute for Mathematics Bonn)
DTSTART:20220513T103000Z
DTEND:20220513T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /13/">Almost optimal Diophantine exponent for $\\mathrm{SL}(n)$</a>\nby Su
 bhajit 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 o
 f $\\mathrm{SL}_n(\\mathbb{Z}[1/p])$ in $\\mathrm{SL}_n(\\mathbb{R})$ in a
  quantitative manner along the line of work by Ghosh--Gorodnik--Nevo. The 
 Diophantine exponent $\\kappa$ for a pair of elements $x\,y \\in\n\\mathrm
 {SL}_n(\\mathbb{R})$ is a certain positive real number that\, loosely\, me
 asures the complexity of an element $\\gamma\\in\\mathrm{SL}_n(\\mathbb{Z}
 [1/p])$ such that $\\gamma x$ approximates $y$ with a prescribed error. Gh
 osh--Gorodnik--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
 \ntechniques 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 t
 he Eisenstein series on reductive groups. This is one of the inputs in our
  proof towards improved Diophantine exponent. This is a joint work with Am
 itay Kamber.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (University of Göttingen)
DTSTART:20220520T103000Z
DTEND:20220520T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/14
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /14/">Density of rational points near/on compact manifolds with certain cu
 rvature conditions</a>\nby Damaris Schindler (University of Göttingen) as
  part of Number theory by the sea\n\n\nAbstract\nIn this talk I will discu
 ss joint work with Shuntaro Yamagishi where we\nestablish an asymptotic fo
 rmula for the number of rational points\, with bounded\ndenominators\, wit
 hin a given distance to a compact submanifold M of R^n with a\ncertain cur
 vature condition. Technically we build on work of Huang on the density of\
 nrational points near hypersurfaces. One of our goals is to explore genera
 lisations\nto higher codimension. In particular we show that assuming cert
 ain curvature\nconditions in codimension at least two\, leads to upper bou
 nds for the number of\nrational points on M which are even stronger than w
 hat would be predicted by the\nanalogue of Serre's dimension growth conjec
 ture.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART:20220603T103000Z
DTEND:20220603T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/15
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /15/">Additive equations over lattice points on spheres</a>\nby Akshat Mud
 gal (University of Oxford) as part of Number theory by the sea\n\n\nAbstra
 ct\nIn this talk\, we will consider additive properties of lattice points 
 on\nspheres. 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 nu
 mber 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 c
 losely related to various problems in harmonic analysis and analytic numbe
 r theory\, including Bourgain's discrete restriction conjecture for sphere
 s. We will survey some recent results in this direction\, as well as descr
 ibe some of the various\ntechniques\, arising from areas such as incidence
  geometry\, analytic number theory\nand arithmetic combinatorics\, that ha
 ve been employed to tackle this type of\nproblem.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Ostafe (The University of New South Wales)
DTSTART:20220610T103000Z
DTEND:20220610T113000Z
DTSTAMP:20260404T094833Z
UID:ntsea/16
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/ntsea
 /16/">Integer matrices with a given characteristic polynomial and multipli
 cative dependence of matrices</a>\nby Alina Ostafe (The University of New 
 South Wales) as part of Number theory by the sea\n\n\nAbstract\nWe conside
 r the set $\\mathcal{M}_n(\\mathbb{Z}\; H))$ of $n\\times n$-matrices with
 \ninteger elements of size at most $H$ and obtain upper and lower bounds o
 n the number\nof $s$-tuples\nof matrices from $\\mathcal{M}_n(\\mathbb{Z}\
 ; H)$\, satisfying various multiplicative\nrelations\, including\nmultipli
 cative dependence\, commutativity and\nbounded generation of a subgroup of
  $\\mathrm{GL}_n(\\mathbb{Q})$. These problems\ngeneralise those studied\n
 in 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-commuta
 tivity 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\nrespect to $f$. This complements\nthe asymptotic formula of A. Eski
 n\, S. Mozes and N. Shah (1996) in which $f$ has to\nbe fixed and irreduci
 ble.\n\nJoint work with Igor Shparlinski.\n
LOCATION:https://stable.researchseminars.org/talk/ntsea/16/
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