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 […]
Title: An intrinsic approach to moduli theoryAbstract: A central problem in algebraic geometry is to construct and study moduli spaces of objects of interesting geometric objects. The classical tool for this is geometric invariant theory, which requires you to approximate a moduli problem by an orbit space X/G for some reductive group G acting on […]
Title: Euler Systems for certain non-unique models Abstract: Pushforwards of Beilinson's Eisenstein symbols into the motives of Shimura varieties provide fertile grounds for exploring instances of the Beilinson–Bloch–Kato conjectures, particularly when these constructions are accompanied by period integral representations. About a decade ago, Pollack and Shah introduced two novel Rankin–Selberg integral representations: one for the […]