# Video Lectures

We introduce new invariants to the existence of Lagrangian

cobordisms in R^4. These are obtained by studying holomorphic disks

with corners on Lagrangian tangles, which are Lagrangian cobordisms

with flat, immersed boundaries.

We...

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level...

Let G be a simply-connected complex semisimple algebraic group and let C be a smooth projective curve of any genus. Then, the moduli space of semistable G-bundles on C admits so called determinant line bundles. E. Verlinde conjectured a remarkable...

There is a celebrated connection between minimal (or constant mean curvature) hypersurfaces and Ricci curvature in Riemannian Geometry, often boiling down to the presence of a Ricci term in the second variation formula for the area. The first goal...

A famous conjecture of Littlewood states that the Fourier transform of every set of N integers has l^1 norm at least log(N), up to a constant multiplicative factor. This was proved independently by McGehee-Pigno-Smith and Konyagin in the 1980s. This...

Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk...

A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. Equivalently, a meandric system is a coupled...

A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of the alternating group An be?

In the talk we will completely solve the problem by...

Approximate lattices in locally compact groups are approximate subgroups that are discrete and have finite co-volume. They provide natural examples of objects at the intersection of algebraic groups, ergodic theory and additive combinatorics... with...

Work of Mark Shusterman and myself has proven an analogue of Chowla's conjecture for polynomial rings over finite fields, which controls k-points correlations of the Möbius function for k bounded by a certain function of the finite field size...