The class number formula describes the behavior of the Dedekind zeta function at s = 0. The Stark conjecture extends the class number formula, describing the behavior of Artin L-functions at s = 0 in terms of units. The Harrisâ??Venkatesh conjecture, originally motivated by the conjectures of Venkatesh and Prasannaâ??Venkatesh on derived Hecke algebras, can be viewed as an analogue to the Stark conjecture modulo p. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harrisâ??Venkatesh and Stark for modular forms of weight 1, and describe the proof of this in the imaginary dihedral case. Time permitting, I will also describe some new questions and in-progress work modulo pn.

In this talk, I will present an overview of the framework originally proposed by Harder and further developed in collaboration with Raghuram to address rationality problems for special values of certain automorphic L-functions. I will then proceed to state my main results on the rationality of the special values of Langlands-Shahidi L-functions appearing in the constant term of the Eisenstein series associated with the exceptional group of type G_2â?? over a totally imaginary number field. This study marks the first instance where rank-one Eisenstein cohomology is employed to investigate the arithmetic of automorphic L-functions in the presence of multiple L-functions.

In this talk I will describe my recent work with Mahesh Kakde on the Brumer-Stark Conjecture and certain refinements. I will give a broad overview that motivates the conjecture and gives connections to explicit class field theory. I will conclude with a description of recent work (joint w/ Kakde, Jesse Silliman, and Jiuya Wang) in which we complete the proof of the conjecture. Moreover, we deduce a certain special case of the Equivariant Tamagawa Number Conjecture, which has important corollaries. The key aspect of the most recent results, which allows us to handle the prime p=2, is the proof of a version of Ribet's Lemma in the case of characters that are congruent modulo p.

In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.

Hilbert modular surfaces are 2-dimensional analogues of modular curves, parametrizing polarized abelian surfaces with endomorphism and level structure. Modular curves are stratified by genus, and canonical equations for modular curves are obtained from the graded ring of modular forms. Similar to how curves are stratified by genus, surfaces are organized by their numerical invariants; the Enriques-Kodaira classification organizes smooth surfaces by Kodaira dimension, Hodge numbers, and Chern numbers. In this talk, we explain how to compute these invariants and equations for certain Hilbert modular surfaces. This is joint work with Eran Assaf, Angie Babei, Ben Breen, Sara Chari, Edgar Costa, Juanita Duque-Rosero, Alex Horawa, Jean Kieffer, Avi Kulkarni, Grant Molnar, Abhijit S. Mudigonda, Michael Musty, Sam Schiavone, Shikhin Sethi, and Samuel Tripp.

The classical branching rules describe the spectrum of an irreducible complex representation of a compact Lie group to its subgroup. The local Ganâ??Grossâ??Prasad conjecture generalizes the branching problem to classical groups over local fields of characteristic zero. After the pioneering work of Waldspurger, there has been significant progress on the conjecture using various approaches. In my talk, I will introduce a relatively uniform approach to prove the conjecture, including joint work with Z. Luo and joint work with R. Chen and J. Zou.

We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which the parameter factors and prove that this version of the correspondence follows if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This is joint work with David Schwein (Bonn).

The module theory of the Weyl algebra, known as the theory of $D$-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic $D$-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties.

With Yongyuan Huang and Jun Bo Lau, we recently completed a census of genus-6 curves over the field F_2, and are working on a similar census in genus 7. This uses Mukai's "flowcharts" for describing canonical curves in this genera. We discuss some of the key features of this classification; some aspects of computational group theory required to convert this classification into tractable computations; and some applications of the results, including relative class number problems for function fields, gonality of curves over finite fields (work of Faber-Grantham-Howe), and cohomology of modular curves (work of Canning-Larson and Bergstrom-Canning-Petersen-Schmitt).