# Colloquium on "Topology of 'manifolds' in positive characteristics" by Lance Miller (University of Arkansas)

Event Type:

Colloquium

Speaker:

Lance Miller (University of Arkansas)

Event Date:

Monday, March 18, 2019 -

3:30pm to 4:30pm

Location:

SMLC 356

Audience:

General Public

Sponsor/s:

Alex Buium

### Event Description:

Abstract:

There is a wealth of topological and analytic tools available for studying complex manifolds. Specifically, one has singular and de Rham cohomology theories, integration of vector fields, L^2-methods, etc. All of these fail for 'manifolds' in positive characteristic, specifically smooth projective varieties over positive characteristic

fields. In this talk, no algebraic geometry will be assumed. I will discuss the difficulties of working with de Rham cohomology of such spaces by surveying work of Grothendieck, Berthelot, and Illusie which helps describe what topological information is present. In particular, I will describe an overview of a purely algebraic approach to get at hidden topological information. Among the most interesting is a cohomology theory described by a de Rham type complex called the de Rham-Witt complex; introduced originally by Illusie. I will give some recent results and applications of the study of this complex and mention some recent joint work with Veronika Ertl.