# School of Mathematics

## Recent progress on Overdetermined Elliptic Problems

In this talk we will survey recent progress on the Beresticky-Caffarelli-Nirenberg Conjecture in Space Forms; that is, let $\Omega$ be an open connected domain of a complete connected Riemannian manifold ($M,g$) and consider the OEP given by

## On the NP-hardness of 2-to-2 Games

The Unique-Games Conjecture is a central open problem in the field of PCP’s (Probabilistically Checkable Proofs) and hardness of approximation, implying tight inapproximability results for wide class of optimization problems.

We will discuss PCPs, the Unique-Games Conjecture and some recent progress. (no familiarity with PCPs or with last week's talk are needed).

## X-Ramanujan graphs: ex uno plures

## 2-universality of random graphs.

## The Zilber-Pink conjecture

The Zilber-Pink conjecture is a far reaching finiteness conjecture in diophantine geometry, unifying and extending Mordell-Lang and Andre-Oort. This lecture will state the conjecture, illustrate its varied faces, and indicate how the point-counting strategy can be applied to parts of it.

## Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

## O-minimality and Ax-Schanuel properties

This lecture will describe the historical context and some key properties of o-minimality. It will then describe certain results in functional transcendence, generalizing the classical results on exponentiation due to Ax, and sketch how they can be proved.

## Existence of infinitely many minimal hypersurfaces in closed manifolds

## Point-counting and diophantine applications

This introductory lecture will describe results about counting rational points on certain non-algebraic sets and sketch how they can be used to attack certain problems in diophantine geometry and functional transcendence.