Title: The Geometry of the Hitchin Fibration for Symmetric Spaces Abstract: Motivated by questions in relative Langlands and relative endoscopy, we study generalized Hitchin systems associated to symmetric spaces. We will use two new techniques in invariant theory, the regular quotient and the parabolic cameral cover, in an effort to describe the geometry of such […]
Title: Why do cusp forms exist? Abstract: I will begin this talk by reviewing the Eichler-Shimura isomorphism between the space of cusp forms of weight k and level N, and a certain cohomology group. Shimura called this cohomology group as parabolic cohomology. In the context of automorphic forms on higher groups, such a cohomology group takes the form […]
Title: Selmer groups, p-adic L-functions and Euler Systems: A new Framework Abstract: Selmer groups are key invariants attached to p-adic Galois representations. The Bloch—Kato conjecture predicts a precise relationship between the size of certain Selmer groups and the leading term of the L-function of the Galois representation under consideration. In particular, when the L-function does […]
Title: Restricted Arithmetic Quantum Unique ErgodicityAbstract: The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I […]
Title: Arithmetic Quantum Field Theory? Abstract: Mathematical structures suggested by quantum field theory have revolutionised important areas of algebraic geometry, differential geometry, as well as topology in the last three decades. This talk will introduce a few of the recent ideas for applying structures inspired by physics to arithmetic geometry.
Title: Compatibility of canonical ell-adic local systems on Shimura varieties Abstract: In his 1979 Corvallis paper, Deligne uses a modular interpretation to construct canonical models of Shimura varieties of Hodge type, and suggested a dream that all Shimura varieties with rational weight should be moduli spaces of motives over number fields. Outside of the abelian […]
Title: The Geometry of Theta Correspondence for Principal Series over Finite Fields Abstract: In this talk, we explore the finite field theta correspondence between principal series representations. We begin by categorifying the relevant Hecke algebra bimodules associated with this correspondence. Utilizing characteristic cycle maps, we provide a detailed description of the theta correspondence for principal […]
Title: The integral converse theorem and no-shadowing bounds for polynomials and L-functions Abstract: I will outline an alternative argument for the complex-analytic ingredient of the "unbounded denominators theorem" joint with Frank Calegari and Yunqing Tang, and then a refinement of the result as a converse theorem for Dirichlet series with almost-integer coefficients. This is followed […]
Title: Torsion vanishing for some Shimura varietiesAbstract: We will discuss joint work with Linus Hamann on generalizing the torsion-vanishing results of Caraiani-Scholze and Koshikawa for the cohomology of Shimura varieties. We do this by applying various geometric methods to understand sheaves on Bun_G, the moduli stack of G-bundles on the Fargues-Fontaine curve. Our method showcases that […]
Title: Twisted Gan-Gross-Prasad conjecture and arithmetic fundamental lemma Abstract: The twisted Gan-Gross-Prasad (GGP) conjecture opens a way of studying (a twist of) central Asai L-values via descents and period integrals. Firstly, I will prove new cases of twisted GGP conjecture (joint work with Weixiao Lu and Danielle Wang), based on the relative trace formula approach […]
Title: On an application of the residue method. Abstract: When a period integral of an automorphic form converges and is not zero it is often related to a special value of an L-function and indicates that the automorphic form is a functorial transfer. When convergence fails it is still interesting to make sense of the […]