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 […]
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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 […]
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