Saturday, November 15, at the University of Maryland

Murilo Corato Zanarella (JHU)

Title: Bipartite Euler systems and local harmonic analysis.
Abstract: In the early 2000’s, Bertolini and Darmon introduced the technique of “bipartite Euler systems” to bound Selmer groups of elliptic curves via level raising congruences. In this talk, I will present a generalization of their method to Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I will highlight a key difficulty that arises due to the lack of local multiplicity one in this unitary Friedberg–Jacquet setting, and the local harmonic analysis used to resolve it.

Zengrui Han (UMD)

Title: Resolution of toric substacks by line bundles revisited.
Abstract: Beilinson’s locally free resolution of the diagonal for the projective space P^n plays an important role in the study of the derived category D^b(P^n). Recently, inspired by homological mirror symmetry for toric varieties, Hanlon, Hicks and Lazarev constructed cellular resolutions of structure sheaves of toric substacks by certain line bundles on the ambient smooth toric stacks, which can be seen as a toric generalization of Beilinson’s construction. In this talk, I will present a more streamlined formulation of their result and a substantial simplification of the proof. If time permits, I will also talk about how to generalize their result to singular toric stacks. This is joint work with Lev Borisov.

Qihang Li (UMD)

Title: Local models and nearby cycles for pro-p Iwahori level.
Abstract: he theory of integral and local models provides powerful tools for studying Shimura varieties with parahoric level structure. Much less, however, is known when the level structure is deeper than the Iwahori level. In this talk, I will review the construction of integral and local models for Siegel-type PEL Shimura varieties with Iwahori level structure, and discuss their generalization to the pro-p Iwahori setting using the theory of Oort–Tate generators of finite flat group schemes of order p. I will also explain how Gaitsgory’s central functor can be adapted to the pro-p Iwahori context and how this yields the central functions predicted by the test function conjecture of Haines and Kottwitz. This talk is based in part on joint work in progress with Thomas Haines and Benoît Stroh.

Javier Reyes (UMD)

Title: Alternative approach to the classification of singular fibers in fibrations.
Abstract: Let X be a smooth surface over the disc D fibered on genus g curves with a singular central fiber. Assuming the fibration is minimal, this is equivalent to giving a map from the punctured disc to the moduli space Mg, and we can determine the structure of the singular fiber from this classifying map. We can use this method to recover the Kodaira-Néron classification of singular elliptic fibers, and give a strategy for an effective classification of singular fibers in hyperelliptic fibrations.

Akira Tominaga (JHU)

Title: Topological Jacobi Forms.
Abstract: The Witten genus forms a bridge between topology and number theory: Witten showed that the elliptic genus of a string manifold takes values in modular forms, and the work of Goerss–Hopkins–Miller identifies it as a ring homomorphism from the bordism ring of string manifolds to the spectrum topological modular forms (TMF). Analogously, Ando–French–Ganter constructed the two-variable elliptic genus, regarded as an “adjoint” of the Witten genus, which assigns to a Calabi–Yau manifold a Jacobi form. In this talk, I will introduce the spectrum of topological Jacobi forms, constructed using the S^1-equivariant TMF developed by Lurie and Gepner–Meier, and discuss its basic properties and its applications.

Yueqiao Wu (JHU)

Title: Valuations at infinity.
Abstract: The interplay between the existence of canonical metrics and the space of valuations on a smooth complex projective variety has been extensively studied over the past decade. Building on this philosophy, it is also understood that if X is a smooth complex affine variety, then certain classes of canonical metrics are closely related to valuations which do not admit a center on X. In this talk, I will explain what these valuations are, and some attempts in studying the algebro-geometric aspects of these valuations. This is based on joint work with Mattias Jonsson.