# School of Mathematics

## Drinfeld's lemma for schemes

## Analyticity results for the Navier-Stokes Equations

## Upper bounds for constant slope p-adic families of modular forms

## A Regularity Lemma with Modifications

Given an arbitrary graph, we show that if we are allowed to modify (say) 1% of the edges then it is possible to obtain a much smaller regular partition than in Szemeredi's original proof of the regularity lemma. Moreover, we show that it is impossible to improve upon the bound we obtain.

## The systole of large genus minimal surfaces in positive Ricci curvature

We prove that the systole (or more generally, any k-th

homology systole) of a minimal surface in an ambient three manifold of

positive Ricci curvature tends to zero as the genus of the minimal

surfaces becomes unbounded. This is joint work with Anna Siffert.

## Minimal hypersurfaces in manifolds of finite volume

## Vortex equation and gauged sigma model

## 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,

## 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.