Saturday, April, 2023 at Johns Hopkins University

Ben Elias (Oregon)

Title: Categorical diagonalization.
Abstract: We give an introduction to the circle of ideas known as categorification, guided by the following question: what might it mean to diagonalize a functor?

Given a category, one can forget most of the information and just remember skeletal information about the objects up to isomorphism, a process known as decategorification. For example, the decategorification of the category of vector spaces is the natural numbers, where a vector space is only remembered by its dimension. What one forgets in this process is the rich structure of morphisms between objects. Meanwhile, categorification is the art of taking something you know and love, and realizing that it was secretly, all along, the decategorification of an interesting category! In this talk, we discuss the case of an invertible functor from a category to itself, which corresponds to a diagonalizable operator acting on the decategorification. What rich structure has been forgotten, and how do we put it back in? We demonstrate examples in the category of modules over the ring Z[x]/(x^2-1). We also wave our hands at some important examples in projective algebraic geometry.

Lillian Pierce (Duke)

Title: Number-theoretic methods to produce counterexamples for questions motivated by PDE’s.
Abstract: In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated question was recently resolved by a combination of two results: one by Bourgain, whose counterexample construction proved a necessary condition, and later a complementary result of Du and Zhang, who proved a sufficient condition.

Bourgain’s counterexample was particularly interesting for two reasons: first, it generated a necessary condition that contradicted what everyone had expected, and second, it was a number-theoretic argument. In this talk we will describe how number theory plays a role, first in the context of Bourgain’s counterexample. Then we will describe a new, far more flexible number-theoretic method for constructing counterexamples, which opens the door to studying convergence questions for other dispersive PDE’s, where many questions remain open. Along the way we’ll see why no mathematics we learn is ever wasted, and how the boundary from one mathematical area to another is not always clear. While the motivation for the problems we describe will come from PDE’s, the talk will not assume specialist knowledge in analysis, and will be aimed for students and researchers in algebra and number theory.

Sug Woo Shin (Berkeley)

Title: Automorphic Galois representations for classical groups
Abstract: It is a theorem due to many people, most recently by Harris-Lan-Taylor-Thorne and Scholze, that there exist Galois representations associated with regular cuspidal automorphic representations of GL(n) over totally real or CM fields. This may be thought of as one direction of the global Langlands correspondence for GL(n). I will explain what we need, especially from the theory of automorphic forms, to go from there to obtain similar results for classical groups. Some open questions will also be discussed.