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 […]
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, […]
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 […]
Title: Bi-ordinary modular forms Abstract: Hida theory provides a p-adic interpolation of modular forms that have an ordinary property. Correspondingly, the p-adic 2-dimensional Galois representations associated to the eigensystems of the Hecke action on these ordinary modular forms have a property known as ordinary: when restricted to a decomposition group at p, this representation is […]
Title and abstract TBA
Title: Complexity of actions over perfect fields Abstract: Let K be a field, let G be a reductive K-group and let X be a G-variety. When K is algebraically closed, Brion and Vinberg proved independently that a Borel subgroup of G has finitely many orbits in X iff it has an open orbit (i.e. X […]
Title: Review of geometric local Langlands for GL1 in mixed characteristic and an introduction to Banach Colmez Spaces. Reference: Hansen's Beijing Lecture Notes on Categorical Local Langlands & Fargues-Scholze.
Title: Positive scalar curvature and exotic structures on simply connected four manifolds.Abstract: We address Gromov’s band width inequality and Rosenberg’s S1-stability conjecture for smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless, we show that both of these results hold upon considering simply […]