# Video Lectures

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the...

Let G be a compact Lie group acting on a closed manifold M. Partially motivated by work of Uhlenbeck (1976), we explore the generic properties of Laplace eigenfunctions associated to G-invariant metrics on M. We find that, in the case where 𝕋 is a...

I will discuss escape of mass estimates for SL(d,ℝ)-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's...

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed...

Agreement testing (aka direct product testing), checks if consistent local information reveals global structure. Beyond its theoretical connections to probabilistic checkable proofs (PCPs), constructing agreement testers is a fundamental...

A deep result of Furstenberg from 1967 states that if Γ is a lattice in a semisimple Lie group G, then there exists a measure on Γ

with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of G

is absolutely...

A dictionary data structure maintains a set of at most n� keys from the universe [U][�] under key insertions and deletions, such that given a query x∈[U]�∈[�], it returns if x� is in the set. Some variants also store values associated to the keys...

For a compact Lie group G and a Hamiltonian G-space M, can we find a smooth weak deformation retraction from a neighbourhood of the zero level set of the momentum map onto it? If we do not require smoothness then this is already known, in fact one...

The symplectic squeezings in the cotangent bundle of a torus is distinct from those in $R^{2n}$, due to the nontrivial topology of the torus. In this talk, we will show that for $n\ge2$ any bounded domain of $T^*T^n$ can be symplectically embedded...

We introduce an equivariant Lagrangian Floer theory on compact symplectic toric manifolds. We define a spectral sequence to compute the equivariant Floer cohomology. We show that the set of pairs $(L,b)$, each consisting of a Lagrangian torus fiber...