# Brandeis Dynamics and Number Theory Seminar 2019-2020

## Seminar Information

### Description

The Brandeis Dynamics and Number Theory Seminar is a research seminar broadly showcasing modern research in ergodic theory and dynamical systems, Lie theory, representation theory, geometry, and their interactions with number theory.

### Time and Location

Thursdays at 3:00PM in Goldsmith 226

## Schedule of Talks

### Fall 2019

September 12Dmitry KleinbockBrandeis UniversityShrinking targets in dynamics and number theoryLink to Abstract
September 19Lam PhamBrandeis UniversityExpansion spectral gaps in Lie groupsLink to Abstract
September 26Dmitry KleinbockBrandeis UniversityShrinking targets in number theory and dynamicsLink to Abstract
October 10Lam PhamBrandeis UniversityClassical results around Kazhdan’s Property $(T)$Link to Abstract
October 17Samuel EdwardsYale UniversityThe horocycle flow on hyperbolic surfacesLink to Abstract
October 24Mishel SkenderiBrandeis UniversityTBALink to Abstract
October 31Marius LemmHarvard UniversityTBALink to Abstract (Unusual time! joint with Everytopic)
October 31Rahul KrishnaBrandeis UniversityTBALink to Abstract
November 7Byungchul ChaMuhlenberg CollegeIntrinsic Diophantine Approximation on SpheresLink to Abstract
November 21Jonathan JaquetteBrandeis UniversityWright’s Conjecture and the Prime Number TheoremLink to Abstract

## Detailed Abstracts

### September 12 @ 3:00PM – Dmitry Kleinbock (Brandeis) – Shrinking targets in dynamics and number theory

Abstract This is going to be an introduction to an active area of ergodic theory and dynamics, with applications to number theory. No background will be assumed, all are welcome.

### September 19 @ 3:00PM – Lam Pham (Brandeis) – Expansion and spectral gaps in Lie groups

Abstract I will survey recent results on various forms of expansion in groups. This is an old topic rooted in Lie theory and representation theory with many deep connections to number theory, geometry, dynamics, and even computing. I will present classical constructions due to Margulis, Kazhdan, and Lubotzky-Phillips-Sarnak and explain the breakthrough work of Bourgain and Gamburd on expansion in thin groups. Finally, I will present related works on improvements and generalizations of the Tits alternative and Kazhdan's Property $(T)$. The talk will be self-contained, with no background assumed.

### September 26 @ 3:00PM – Dmitry Kleinbock (Brandeis) – Shrinking targets in number theory and dynamics

Abstract This is a sequel to the talk I gave on Sept 12. I will provide its brief synopsis (so it's OK if you missed it) and then proceed to describe more slowly what I briefly mentioned at the end of the previous talk, give examples and applications, and highlight the remarkable connection of ergodic theory with shrinking targets to Diophantine approximation.

### October 10 @ 3:00PM – Lam Pham (Brandeis) – Classical reulsts around Kazhdan’s Property $(T)$

Abstract In 1967, Kazhdan introduced (at the time only 20 years old) introduced a fundamental concept in representation theory, now known as Property $(T)$. I will give an introduction to Property $(T)$ with proofs of classical results and some examples and applications. With roots in Lie groups and representation theory, it has been highly influential in several other areas of mathematics, most notably geometry, number theory, dynamics, rigidity, and combinatorics. The talk should be accessible to any graduate student.

### October 17 @ 3:00PM – Samuel Edwards (Yale University) – The horocycle flow on hyperbolic surfaces

Abstract The horocycle flow on finite-volume hyperbolic surfaces is one of the most well-understood unipotent flows in homogeneous dynamics. In particular, its relation with the geodesic flow allows one to use exponential mixing to obtain “polynomially fast” effective equidistribution of all non-closed horocycle orbits. We will discuss a few aspects of the statements of effective equidistribution, and explain how similar results may be obtained for the horocycle flow on infinite-volume geometrically finite hyperbolic surfaces.

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### November 7 @ 3:00PM – Byungchul Cha (Muhlenberg College) – Intrinsic Diophantine Approximation on Spheres

Abstract Let $S^1$ be the unit circle in $\mathbb{R}^2$ 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 part 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 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. In addition, we present similar results for a few different subsets $Z$ of $S^1$. Finally, we report some partial results of similar type for $S^2$. This is joint work with Dong Han Kim

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