# School of Mathematics

## The general case?

## Singular Hodge theory of matroids

## Front propagation in a nonlocal reaction-diffusion equation

We consider a reaction-diffusion equation with a nonlocal reaction term. This PDE arises as a model in evolutionary ecology. We study the regularity properties and asymptotic behavior of its solutions.

## Equivariant and nonequivariant contact homology

I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our construction.

## A Brief Tour of Proof Complexity: Lower Bounds and Open Problems

I will give a tour of some of the key concepts and ideas in proof complexity. First, I will define all standard propositional proof systems using the sequent calculus which gives rise to a clean characterization of proofs as computationally limited two-player games. I will also define algebraic and semi-algebraic systems (SOS, IPS, Polynomial Calculus).

## Gap and index estimates for Yang-Mills connections in 4-d

In this talk I want to discuss two related questions about

## Multiplicity One Conjecture in Min-max theory

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one.

## An Application of the Universality Theorem for Tverberg Partitions

We show that, as a consequence of a remarkable new result of

Attila P\'or on universal Tverberg partitions, any large-enough set

$P$ of points in $\Re^d$ has a $(d+2)$-sized subset whose Radon point

has half-space depth at least $c_d \cdot |P|$, where $c_d \in (0, 1)$

depends only on $d$. We then give an application of this result to

computing weak $\eps$-nets by random sampling. Joint work with Nabil

Mustafa.