## A mountain pass theorem for minimal hypersurfaces with fixed boundary

## One-relator groups, non-positive immersions and coherence

Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups. (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups. But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.

## On the Approximation Resistance of Balanced Linear Threshold Functions

## 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 Party for Which People? The Democrats from Andrew Jackson to Barack Obama and Beyond

## 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).