Saturday, November 23, at Johns Hopkins University

Laura DeMarco (Harvard)

Title: From Manin–Mumford to Dynamical Rigidity.
Abstract:  In the early 1980s, Raynaud proved a theorem (the Manin–Mumford Conjecture) about the geometry of torsion points in abelian varieties, using number-theoretic methods.  Around the same time, and with completely different methods, McMullen proved a dynamical rigidity theorem for holomorphic maps on P¹.  In recent work, joint with Myrto Mavraki, we explained how to view these results as special cases of a unifying conjecture.  (The conjectural statement is inspired by a theorem of Gao and Habegger, called “Relative Manin–Mumford”, and recent results in complex dynamics of Dujardin, Gauthier, Vigny, and others.)  My aim in these two lectures is to present the conjecture with lots of examples and explain what is known, while assuming no background in dynamical systems.  

Alexander Petrov (MIT)

Title: Characteristic classes of p-adic local systems.
Abstract: A useful tool in studying vector bundles on topological spaces or algebraic varieties is characteristic classes in cohomology, such as Chern classes. For vector bundles equipped with a flat connection, Chern classes vanish in cohomology with rational coefficients, but such bundles have a non-trivial theory of secondary characteristic classes, often called Chern-Simons classes, arising from classes in cohomology of the group GL_n(C).
In a joint ongoing project with Lue Pan we study a p-adic analog of this theory. Given an etale Z_p-local system on an algebraic variety, continuous cohomology classes of the group GL_n(Z_p) give rise to natural characteristic classes in the (absolute) etale cohomology of the variety. These classes turn out to vanish in degrees >1 for smooth varieties over algebraically closed fields of characteristic zero (at least for p large as compared to the rank of the local systems). But they are often non-zero for local systems on varieties over non-closed fields and can be partially expressed in terms of Hodge-theoretic invariants of the local system. This is achieved using some relative p-adic Hodge theory, in particular the notion of Chern classes for pro-etale vector bundles, and vector bundles on the Fargues-Fontaine curve.

Mihnea Popa (Harvard)

Title: Hodge symmetries of singular varieties.
Abstract: The Hodge diamond of a smooth projective complex variety contains essential topological and analytic information, including fundamental symmetries provided by Poincaré and Serre duality. I will describe recent progress on understanding how much symmetry there is in the analogous Hodge–Du Bois diamond of a singular variety, and the concrete ways in which this symmetry reflects the singularity types. In the process, we will see how invariants from commutative algebra and birational geometry influence the topology of an algebraic variety, for instance by means of new weak Lefschetz theorems.