# School of Mathematics

## Weyl Law for the phase transition spectrum and density of limit-interfaces

Abstract: The Allen-Cahn equation behaves as a desingularization of the area functional. This allows for a PDE approach to the construction of minimal hypersurfaces in closed Riemannian manifolds. After presenting and overview of the subject, I will discuss recent results regarding a Weyl Law and its consequences for the density of minimal hypersurfaces in generic metrics. This is joint work with P. Gaspar.

## Some minimal submanifolds generalizing the Clifford torus

Abstract: The Clifford torus is the simplest nontotally geodesic minimal surface in S^3. It is a product surface, it is helicoidal, and it is a solution obtained by separation of variables. We will show that there are more minimal submanifolds with these properties in S^n and in R^4.

## Analysis of some Conformally Invariant Problems

## 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).

## 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

## 2-universality of random graphs.

## X-Ramanujan graphs: ex uno plures

## 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.