Title: Reconstructing schemes from their étale topoi.Abstract: In Grothendieck’s 1983 letter to Faltings that initiated the study of anabelian geometry, he conjectured that a large class of schemes can be reconstructed from their étale topoi. In this talk, I’ll discuss joint work with Magnus Carlson and Sebastian Wolf that proves Grothendieck’s conjecture for infinite fields. Specifically, we […]
Title: Towards Bezrukavnikov's geometrization of affine Hecke algebras in mixed-characteristics Abstract: Let G be a connected reductive group over a p-adic field. Kazhdan and Lusztig established an isomorphism between the extended affine Hecke algebra of G and certain equivariant K-group of the Steinberg variety of the Langlands dual group of G. This isomorphism plays a crucial […]
Title and abstract TBA
Title: Transfer operators for the relative trace formula. (Cont.) Abstract: Last time I presented an explicit formula for transfer operators that relate test measures and characters between the double quotients HG/H, for any rank-one homogeneous space G/H and the Whittaker model of SL(2) or PGL(2). In this talk, I will give a possible explanation for […]
Title: Restriction estimates for spectral projectionsAbstract: Restriction estimates for spectral projections have been widely studied since the work of Burq, Gérard, and Tzvetkov as a method for investigating eigenfunction concentration. The problem of establishing the optimal $L^p$ bounds for the restriction of Laplace-Beltrami eigenfunctions remains open, particularly when the restriction submanifold is of codimension 1 […]
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Title: Beyond Endoscopy via Poisson Summation for $GL(2,K)$ Abstract: Langlands proposed a strategy called Beyond Endoscopy to prove the principle of functoriality, which is one of the central questions of present day mathematics. Langlands strategy of beyond endoscopy is a two-step process where the first step isolates the packets of cuspidal automorphic representations whose $L$-functions […]
Title and abstract TBA
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 […]