# Quantitative ell-adic sheaf theory

## Location

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.