# School of Mathematics

## On the topology and index of minimal surfaces

## Non-commutative rank

A linear matrix is a matrix whose entries are linear forms in some indeterminates $t_1,\dots, t_m$ with coefficients in some field $F$. The *commutative rank* of a linear matrix is obtained by interpreting it as a matrix with entries in the function field $F(t_1,\dots,t_m)$, and is directly related to the central PIT (polynomial identity testing) problem. The

## Spacetime positive mass theorem

## Near-Optimal Strong Dispersers

Randomness dispersers are an important tool in the theory of pseudorandomness, with numerous applications. In this talk, we will consider one-bit strong dispersers and show their connection to erasure list-decodable codes and Ramsey graphs.

## The Sample Complexity of Multi-Reference Alignment

## 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.