The integral cohomology groups of a complex algebraic variety are one of the most fundamental invariants associated to the variety. The ranks of these groups are well understood in terms of the equations defining the variety, thanks to Hodge theory. However, the torsion tends to be much more "transcendental" in nature and not easily accessible via algebraic techniques. Torsion cohomology classes have recently played a pivotal role in fundamental advances in many subjects such as number theory, algebraic geometry, and representation theory, so it is important to better understand torsion from an algebraic perspective.

In my talk, I'll explain how to bound the torsion explicitly in terms of the equations defining the variety. The bound is a consequence of the construction of a new cohomology theory in p-adic Hodge theory, and the bulk of my talk will be dedicated to explaining why the coefficient ring of this cohomology theory (i.e., its value on a point) makes meaningful mathematical sense of a small piece of the non-existent object "Z tensor Z over F_1".

This talk is based on joint work with Matthew Morrow and Peter Scholze. Speaker(s): Bhargav Bhatt (University of Michigan)