# School of Mathematics

## 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.

## Small-Set Expansion on the Grassmann Graph.

This talk is mainly concerned with the investigation of small set expansion on the Grassmann Graphs, a study that was motivated by recent applications to Probabilistically Checkable Proofs and hardness of approximation.