Fall 2022 Titles and Abstracts

Date: 22 September 2022
Speaker: Simon Baker
Title: Overlapping iterated function systems from the perspective of Metric Number Theory
Abstract: Khintchine’s theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the divergence of naturally occurring volume sums. Importantly this result provides a quantitative description of how the rationals are distributed within the 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 systems these results apply to include those arising from Bernoulli convolutions, the 0,1,3 problem, and affine contractions with varying translation parameters. 
Time permitting I also will discuss a particular family of iterated function systems for which we can be more precise. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.

Date: 6 October 2022
Speaker: Juno Seong
Title: An avoidance principle and Margulis functions for expanding translates of unipotent orbits
Abstract:  Avoidance principles — quantifying how much time trajectories avoid certain subsets of the ambient space — have been fruitful in the study of dynamical 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 upper 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 joint work with Anthony Sanchez.

Date: 13 October 2022
Speaker: Christopher Lutsko
Title: A Spectral Approach to Counting and Equidistribution
Abstract: Since 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 Gauss circle problem, and the Apollonian counting problem. By surveying results on these problems (and some generalizations), I’ll demonstrate how to use spectral methods to obtain effective asymptotics for some very classical problems. Then I will try and explain how to generalize this method to apply to certain horospherical equidistribution theorems.

Date: 20 October 2022
Speaker: Lam Pham
Title: Short closed geodesics in higher rank arithmetic locally symmetric spaces
Abstract: A well-known conjecture of Margulis predicts that there is a uniform lower bound on the 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 which hold unconditionally and highlight the structure of the bottom of the length spectrum.

Date: 3 November 2022
Speaker: Shreyasi Datta
Title: p-Adic Diophantine approximation with respect to fractal measures
Abstract: In a recent work with Anish Ghosh and Victor Beresnevich we solved a conjecture of
Kleinbock and Tomanov, which shows pushforward of a p-adic fractal measure by ‘nice’
functions exhibits ‘nice’ Diophantine properties. 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, and how we can
sometimes overcome this problem.

Date: 10 November 2022
Speaker: Nikolay Moshchevitin
Title: On inhomogeneous Diophantine approximation
Abstract: We will discuss some classical and modern results related to systems of inhomogeneous linear forms. We  will begin with Kronecker approximation theorem and famous results by Khintchine and continue with rather modern problems, in particular related to weighted setting and coprime approximation.

Date: 17 November 2022
Speaker: Nicolas de Saxce
Title: Rational approximations to linear subspaces
Abstract: Using diagonal orbits on the space of lattices, we revisit some old questions of Schmidt concerning diophantine approximation on Grassmann varieties, and in particular, we prove a version of Dirichlet’s principle in that setting.

Date: 1 December 2022
Speaker: Tariq Osman
Title: Tail Asymptotics for Generalised Theta Sums with Rational Parameters
Abstract: We define generalised theta sums as exponential sums of the form S^f_N(x; \alpha, \beta) := \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 numbers, and x is chosen randomly from the unit interval, we may use homogeneous dynamics to show that N^{-1/2} S^f_N$ possesses a limiting distribution as N goes to infinity, provided f is sufficiently regular. In joint work with F. Cellarosi, we prove that for specific rational pairs (\alpha, \beta) this limiting distribution is compactly supported and that all other rational pairs lead to a limiting distribution with heavy tails. This complements the existing work of F. Cellarosi and J. Marklof where at least one of \alpha or \beta is irrational.

Date: 8 December 2022
Speaker: Sam Chow
Title: Counting rationals and diophantine approximation on fractals
Abstract: We count rationals in missing-digit sets, with applications to diophantine approximation. In the process, we develop the theory of Fourier \ell^1 dimension, including the computational aspect.