Schedule, Notes/Handouts & Videos

I will describe how studying descent for invertible objects gives computational tools for determining invariants such as Picard or local Brauer groups. The illustrative examples will be related to K-theory and topological modular forms. This is based on work in progress with Antieau and Meier.

Hypersurfaces in  projective space cut collections of points on a curve contained in that space. The maximal rank conjecture makes a prediction for the behavior of these collection of points when both the abstract curve and its immersion in projective space are general. A more refined version of the  question asks what happens when the curve is assumed to be general but the map to projective space is not .There is a predicted generic behavior. Moreover, under certain numerical conditions, the failure of that behavior gives rise to a virtual divisor in the moduli space of curves. Understanding of these divisors might allow to show that the moduli spaces of curves of genus 22 and 23 are of general type.

We will compute the p-adic integral etale cohomology of the Drinfeld half-space. Via the integral comparison theorem of Bhatt-Morrow-Scholze this reduces to a computation of the integral de Rham cohomology. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

The talk will describe analytifications and tropicalizations of moduli spaces of algebraic curves and spin curves and their connection. In particular, the moduli space of tropical spin curves will be constructed, and shown to be the tropicalization of the moduli space of algebraic spin curves. The talk will survey recent results obtained jointly with M. Melo and M. Pacini, together with various, less recent, results.