# Quantitative $\ell$-adic sheaf theory

*Will Sawin (Columbia)*

**Fri Jan 20, 20:00-21:00 (3 weeks ago)**

**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.

algebraic geometry

Audience: researchers in the topic

**Stanford algebraic geometry seminar **

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Organizer: | Ravi Vakil* |

*contact for this listing |