**Representation Theory and Number Theory Seminar 2016-17**

Fridays, 3-4pm, LCB 222

** Next Seminar:
**
January 20, **Andrew Snowden (Michigan),**
Integral structures on de Rham cohomology

Abstract:
Given a smooth projective variety X over a number field K, we
construct two canonical O_K-lattices in the algebraic de Rham
cohomology of X. The first is constructed using the p-adic comparison
theorems (for all p). The second is constructed geometrically, but the
proof that it is a lattice uses the p-adic comparison theorems. Both
constructions have more elementary analogs in complex geometry that I
will discuss first. This is joint work with Bhargav Bhatt.

**Full Schedule**:

January 20, Andrew Snowden (Michigan),
Integral structures on de Rham cohomology

Abstract: Given a smooth projective variety X over a number field K, we construct two canonical O_K-lattices in the algebraic de Rham cohomology of X. The first is constructed using the p-adic comparison theorems (for all p). The second is constructed geometrically, but the proof that it is a lattice uses the p-adic comparison theorems. Both constructions have more elementary analogs in complex geometry that I will discuss first. This is joint work with Bhargav Bhatt.

January 27, Brandon Levin (Chicago), The geometry of deformations of Galois representations and
the Breuil-Mézard conjecture.

Abstract: With the introduction of the Taylor-Wiles method in the proof of modularity of elliptic curves over Q, the deformation theory of Galois representations became central to questions in the Langlands program. The most subtle such deformation problem arises when studying p-adic representations of the Galois group of a p-adic field. I will discuss some new methods for studying the geometry of these deformation spaces. I deduce, as a consequence, instances of the Breuil-Mézard conjecture which describes the geometry in terms of representation-theoretic data. This is joint work with Daniel Le, Bao V. Le Hung and Stefano Morra.