Saturday, April 20, at the University of Maryland

Jeremy Hahn (MIT)

Title: Exotic spheres from p-adic cohomology theories
Abstract:  Part 1: An n-dimensional exotic sphere is a smooth manifold homeomorphic, but not diffeomorphic, to S^n. I will give a leisurely introduction to the telescope conjecture in stable homotopy theory (recently resolved in joint work with Burklund, Levy, and Schlank), by explaining its implications for the diffeomorphism classification of exotic spheres.  In the end, understanding the telescope conjecture reduces to a computation in algebraic K-theory.

Part 2: I will explain recent computational developments in algebraic K-theory, which among many other applications lead to new constructions of exotic spheres.  This will be a survey highlighting many scholars’ recent works, but with a focus on connections to cohomology theories of p-adic formal schemes.

David Helm (Imperial)

Title: Finiteness for Hecke algebras of p-adic reductive groups.
Abstract: Part 1: Let F be a p-adic field, and G the F-points of a reductive group over F.  For any compact open subgroup U of F, one can form the Hecke algebra C[U\G/U] of complex valued functions on G, invariant under left and right translation by U.  This space has the structure of a noncommutative ring under convolution.  A celebrated result of Bernstein from the mid-’80s proves that this ring is Noetherian, and in fact proves a stronger statement: that the center of such a Hecke algebra is a finitely generated C-algebra, and that the full Hecke algebra is finitely generated as a module over its center.  We will discuss Bernstein’s proof of this result, and some of its consequences.

Part 2: Bernstein’s result over the complex numbers makes is natural to ask if one still has strong finiteness results for Hecke algebra R[U\G/U], in which the complex coefficients have been replaced by a ring R such as Z_{\ell} or even Z[1/p].  Such questions have natural arithmetic implications, but turn out to be much more difficult, and in full generality this question has remained open from the ’80s until quite recently.  We show how one can use the Fargues-Scholze geometrization of the local Langlands correspondence to finally settle the question.  This is joint work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

Karl Schwede (Utah)

Title: Singularities in Mixed Characteristic via Alterations.
Abstract: Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively. In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of the base and the resolution. In characteristic p > 0, test ideals were originally defined via Frobenius, but under moderate hypotheses, can be computed via a sufficiently large alteration again via canonical modules. In mixed characteristic (for example over the p-adic integers) we show that various mixed characteristic analogs of multiplier/test ideals can be computed via a single sufficiently large alteration, at least when one builds in a small perturbation term. This perturbation is particularly natural from the perspective of almost mathematics, from the theory of tight closure in characteristic p, or the theory of pairs from birational algebraic geometry. Besides unifying the three pictures, this has various applications. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron and Jakub Witaszek.