Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.