Will Sawin, February 10, 2023
Title: Quantitative ell-adic sheaf theory
Abstract: Sheaf cohomology is a powerful tool both in algebraic geometry and its applications to other fields. Often, one wants to prove bounds for the dimension of sheaf cohomology groups. Katz gave bounds for the dimension of the étale cohomology groups of a variety in terms of its defining equations (degree, number of equations, number of variables). But the utility of sheaf cohomology arises less from the ability to compute the cohomology of varieties and more from the toolbox of functors that let us construct new sheaves from old, which we often apply in quite complicated sequences. In joint work with Arthur Forey, Javier Fresán, and Emmanuel Kowalski, we prove bounds for the dimensions of étale cohomology groups which are compatible with the six functors formalism (and other functors besides) in the sense that we define the "complexity" of a sheaf and control how much the complexity can grow when we apply one of these operations.