# School of Mathematics

## Online Learning in Reactive Environments

## Permutation property testing

## How will we do mathematics in 2030 ?

We make the case that over the coming decade, computer assisted reasoning will become far more widely used in the mathematical sciences. This includes interactive and automatic theorem verification, symbolic algebra, and emerging technologies such as formal knowledge repositories, semantic search and intelligent textbooks.

## Thresholds Versus Fractional Expectation-Thresholds

Given an increasing family F in {0,1}^n, its measure according to mu_p increases and often exhibits a threshold behavior, growing quickly as p increases from near 0 to near 1 around a specific value p_c. Thresholds of families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area.

## A rigorous derivation of the kinetic wave equation

In this talk I will outline recent work in collaboration with Pierre Germain, Zaher Hani and Jalal Shatah regarding a rigorous derivation of the kinetic wave equation. The proof presented will rely of methods from PDE, statistical physics and number theory.

## On the gradient-flow structure of multiphase mean curvature flow

Due to its importance in materials science where it models the slow relaxation of grain boundaries, multiphase mean curvature flow has received a lot of attention over the last decades.

## Graph Sparsification via Short Cycle Decomposition

## The h-principle in symplectic geometry

Symplectic geometry, and its close relative contact geometry, are geometries closely tied to complex geometry, smooth topology, and mathematical physics. The h-principle is a general method used for construction of smooth geometric objects satisfying various underdetermined properties. In the symplectic context, h-principles typically give constructions of surprising exotica, and methods for detecting the basic flexible objects. We survey a number of results from the previous decade.

## Topology of resolvent problems

In this talk I will describe a topological approach to some problems about algebraic functions due to Klein and Hilbert. As a sample application of these methods, I will explain the solution to the following problem of Felix Klein: Let $\Phi_{g,n}$ be the algebraic function that assigns to a (principally polarized) abelian variety its $n$-torsion points. What is the minimal $d$ such that, after a rational change of variables, $\Phi_{g,n}$ can be written as an algebraic function of $d$ variables? This is joint work with Mark Kisin and Jesse Wolfson.