# School of Mathematics

## Minimal hypersurfaces in manifolds of finite volume

## PCP and Delegating Computation: A Love Story.

In this talk, I will give an overview on how PCPs, combined with cryptographic tools,

are used to generate succinct and efficiently verifiable proofs for the correctness of computations.

I will focus on constructing (computationally sound) *succinct* proofs that are *non-interactive*

(assuming the existence of public parameters) and are *publicly verifiable*.

In particular, I will focus on a recent result with Omer Paneth and Lisa Yang,

where we show how to construct such proofs for all polynomial time computations,

## (Non)uniqueness questions in mean curvature flow

Mean curvature flow is the negative gradient flow of the

volume functional which decreases the volume of (hyper)surfaces in the

steepest way. Starting from any closed surface, the flow exists

uniquely for a short period of time, but always develops singularities

in finite time. In this talk, we discuss some non-uniqueness problems

of the mean curvature flow passing through singularities. The talk is

mainly prepared for non-specialists of geometric flows.

## Symplectic methods for sharp systolic inequalities

In this talk I would like to explain how methods from

symplectic geometry can be used to obtain sharp systolic inequalities.

I will focus on two applications. The first is the proof of a

conjecture due to Babenko-Balacheff on the local systolic maximality

of the round 2-sphere. The second is the proof of a perturbative

version of Viterbo's conjecture on the systolic ratio of convex energy

levels. If time permits I will also explain how to show that general

systolic inequalities do not exist in contact geometry. Joint work

## New Results on Projections

What is the largest number of projections onto k coordinates guaranteed in every family of m binary vectors of length n? This fundamental question is intimately connected to important topics and results in combinatorics and computer science (Turan number, Sauer-Shelah Lemma, Kahn-Kalai-Linial Theorem, and more), and is wide open for most settings of the parameters. We essentially settle the question for linear k and sub-exponential m.

Based on joint work with Noga Alon and Noam Solomon.

## Regularity of weakly stable codimension 1 CMC varifolds

## Ramanujan complexes and golden gates in PU(3).

In their seminal works from the 80's, Lubotzky, Phillips and Sarnak proved the following two results: