Title: Shimura Varieties and Eigensheaves Abstract: The cohomology of Shimura varieties is a fundamental object of study in algebraic number theory by virtue of the fact that it is the only known geometric realization of the global Langlands correspondence over number fields. Usually, the cohomology is computed through very delicate techniques involving the trace formula. […]

Title: p-adic L-functions for GSp(4)times GL(2) Abstract: I'll explain a construction of p-adic L-functions for GSp(4)times GL(2) by using Furusawa's integral and the proof of its interpolation formula. I'll describe how local functional equations are used to compute the zeta intgerals at p and how the archimedean integrals are computed by using Yoshida lifts together […]

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: 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: 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: 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: 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 […]

Title: Near-center derivatives and arithmetic 1-cycles Abstract: Degrees of arithmetic special cycles on Shimura varieties are expected to appear in first derivatives of automorphic forms and L-functions, such as in the Gross--Zagier formula, Kudla's program, and the Arithmetic Gan--Gross--Prasad program. I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program, which […]

Title: Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on GL_2. Abstract: In the first part of the talk, we will give a direct proof of a Poisson summation formula on the Kuznetsov quotient of GL_2 that is responsible for the local Hankel transform calculated by H. Jacquet. In the second part of the […]

Title: Affine and asymptotic Hecke algebras Abstract: Affine Hecke algebras play a prominent role the representation theory of p-adic groups, where a large subcategory of representations are equivalent to modules over the affine Hecke algebra. The important subcategory of tempered representations is equivalent to modules over a larger ring, the Harish-Chandra Schwartz algebra. However, this […]

Title: Tame Betti local Langlands Abstract: Representations of finite Weyl groups can be realized inside the top cohomology of Springer fibers. By replacing cohomology by equivariant K-theory, Kazhdan and Lusztig classified tamely ramified representations of p-adic groups. Arkhipov and Bezrukavnikov categorified the Kazhdan-Lusztig isomorphism by proving a coherent realization of the affine Hecke category. I […]

Title: Real groups, symmetric varieties, quantum groups, and  Langlands duality Abstract: I will explain a connection between relative Langlands duality and geometric Langlands on real forms of the projective line (i.e. the real projective line or the twistor P1), then explain recent results using this to answer some questions in Langlands duality for real groups and symmetric […]