**Lecture 1: Rigidity properties of higher rank diagonal groups and diophantine approximations**

**Date & Time: **Wednesday, 2 October, 2019, 10:00 hrs

Abstract: In my talk I will present joint work with M. Einsiedler regarding the following natural question: given a number alpha in R, how well can alpha be approximated by a rational p/(q_{1}q_{2}) with q_{1},q_{2}<$\phi$ (Q)? We show that outside a set of Q’s of logarithmic density 0, the trivial bound on how well one can approximate can be improved.

The proof uses a new measure classification theorem.

**Lecture 2: Additive combinatorics and equidistribution**

**Date & Time: **Thursday, 3 October, 2019, 10:00 hrs

Additive combinatorics provide powerful tools to study equidistribution questions quantitatively. I would discuss one instance of this: a result of Bourgain, Furman, Mozes and myself that gives a quantitative equidistribution statement for a random walk on $d$-dimensional torus using matrices in $SL(d, \mathbb{Z})$, as well as some newer results.

**Lecture 3: Dynamics and geometry of numbers**

**Date & Time: **Thursday, October 3, 2019 at 16:00 hrs

Abstract: Somewhat surprisingly, dynamical systems --- a field that can trace its origins to attempts to understand celestial dynamics and other physically evolving systems --- has proved itself to be a powerful method in several areas of number theory, including regarding the study of integer points in natural varieties.

I will focus in particular on two such examples: the triplets $(a,b,c)$ on the two sphere $a^2+b^2+c^2=d$, as well as the integer points on the hyperboloid of one or two sheets (depending on the sign of $d$) where $b^2-4ac=d$.

The study of these Diophantine equations dates quite a long way back: the question of which integers $d$ can be written as a sum of three squares was already answered by Lagrange and Gauss around the turn of the 18th century. Gauss also gave a surprising and deep connection between integer points on a sphere and on the corresponding hyperboloid.

Dynamical ideas were injected to the study of these points by Linnik starting at the end of the 1930s, and his deep insight meshes very well with some recent developments in a subfield of dynamical systems called homogeneous dynamics. I will explain the connection between the study of such integer points and homogeneous dynamics, as well as present some of the new results regarding integer points that were proved using dynamical systems.

This lecture series is also part of the program Smooth and Homogeneous Dynamics.