We welcome everyone!
Organizers: Benjamin Brubaker, Tsao-Hsien Chen, Adrian Diaconu, Haoyang Guo, Dihua Jiang, Kai-Wen Lan
Time: Friday, 11:05 am - 12:35 pm
Location: Vincent Hall 207
Abstract: In 1972, Katz proved the p-curvature conjecture for linear differential equations coming from the cohomology of algebraic varieties. I'll explain how to execute a version of his argument for non-linear differential equations coming from non-Abelian cohomology. This will be a more technical sequel to my colloquium, though there will be no overlap in the results discussed and it will not assume you went to the previous talk.
Abstract: A fundamental problem in arithmetic geometry is the Tate conjecture, which predicts a description of algebraic cycles of a given variety using linear algebraic data. In this talk, we review the basics of the conjecture, including the proof for divisors on abelian varieties and on K3 surfaces, together with the Kuga--Satake correspondence. Then we report on the joint work with Ziquan Yang on the Tate conjecture for minimal surfaces of geometric genus one.
Abstract: In 2001, Conrey, Farmer, Keating, Rubinstein, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of L-functions. This heuristic was later extended by Andrade and Keating to include moments and ratios of the family of L-functions associated to hyperelliptic curves of genus g over a fixed finite field.
In joint work with Bergström, Petersen, and Westerland, we related the moment conjecture of Andrade and Keating to the problem of understanding the homology of the braid group with symplectic coefficients. We computed the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations, and showed that the answer matches the number-theoretic predictions. Our results, combined with a recent homological stability theorem of Miller, Patzt, Petersen, and Randal-Williams, imply the conjectured asymptotics for all moments in the function field case, for all large enough odd prime powers q.
Abstract: The relative Langlands program of Ben-Zvi–Sakalleridis-Venkatesh offers a framework for describing the category of sheaves on the loop space LX of a G-variety X, where G is a connected reductive algebraic group. Their conjectures generalize the celebrated geometric Satake equivalence of Ginzburg and Mirković–Vilonen. I will survey their ideas and, building on work of Ginzburg–Riche, describe the case of X = G/L, where L is a Levi subgroup of G.
Abstract: According to the relative Langlands functoriality conjecture, an admissible morphism between the L-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the fundamental lemma and the local transfer on the geometric side of the relative trace formulas. In this talk, we will consider the rank-one spherical variety case, where the admissible morphism between the L-groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer. If time permits, we will also talk about some global aspects.