The Zilber-Pink conjecture

Jonathan Pila
University of Oxford
October 26, 2018

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

Rong Zhou
Member, School of Mathematics
October 25, 2018
Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group.

Small-Set Expansion on the Grassmann Graph.

Dor Minzer
Member, School of Mathematics
October 23, 2018
A graph G is called a small set expander if any small set of vertices contains only a small fraction of the edges adjacent to it.
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.