Hida theory provides a p-adic interpolation of modular forms that have a property known as ordinary. Hida’s interpolation inspired Mazur to formulate the deformation theory of Galois representations, which Wiles used to prove (among other things) that p-adically interpolated modular form correspond closely, via an “R=T theorem”, to p-adically deformed ordinary 2-dimensional Galois representations. This notion of “ordinary” 2-dimensional Galois representation means that the representation becomes reducible when restricted to a decomposition subgroup at the prime p. But which ordinary modular forms have a Galois representation that is not only reducible at p but also decomposable at p? After explaining some arithmetic-geometric motivations for this question, I will explain some joint work with Francesc Castella in which we construct a length 1 “bi-ordinary complex” of modular forms that has a Hida-type interpolation property and whose associated Galois representations are reducible decomposable at p. This construction builds upon on Coleman’s work on presenting de Rham cohomology of modular curves as a quotient of differentials of the second kind, as well as Boxer and Pilloni’s work on higher Hida theory.

Gravitational instantons are non-compact Calabi--Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe three projects on gravitational instantons including: (a) Fredholm theory and deformation of the ALH* type; (b) non-collapsing degeneration limits of ALH* and ALH types; (c) existence of stable non-holomorphic minimal spheres in some ALF types. These three projects utilize geometric microlocal analysis in different singular settings. Based on works joint with Rafe Mazzeo, Yu-Shen Lin and Sidharth Soundararajan.

This talk walks through how we have addressed some emerging challenges of the SARS-CoV-2 pandemic from a mathematical perspective. Of initial importance was the accurate classification of samples as positive or negative, which is difficult when the corresponding measurement values overlap. We separated populations by using more available measurements per person to build probabilistic models that capture structural characteristics of the data. These models are inputs to a framework for minimal-error classification. As vaccines were introduced, we extended this classification to situations with more than two classes. Such a multiclass classification is complicated when the relative prevalence in each class is unknown; we designed a prevalence estimation method that is independent of classification. Of interest now is the time-dependent viral response to infections and vaccinations. In recent work, we developed a Markov chain model for state transitions coupled with a probabilistic framework for post-infection or post-vaccination antibody kinetics. This is an important step towards a comprehensive understanding of antibody kinetics that could allow us to analyze the protective power of natural immunity or vaccination and provide booster timing recommendations. Our work involves collaboration with epidemiologists and immunologists.

Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.

I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.

I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song and Tian posed several conjectures which describe this behavior. I will report on recent work (joint with Hein and Lee) which confirms these conjectures.

Homogenization is the approximation of a complex, “disordered” system by a simpler, “ordered” one. Picture a walker on a grid. In each step, the walker chooses to walk along a neighboring edge with equal probability. At large scales, the walker approximates Brownian motion. But what if some edges are more likely to be traversed than others? I will discuss recent advances in the theory of quantitative homogenization which make it possible to analyze random walk with drift and other models in probability. Joint work with Scott Armstrong and Tuomo Kuusi.

The categorical p-adic local Langlands correspondence has been established for the group GL_2(Q_p) in joint work of the speaker with Andrea Dotto and Toby Gee. In this talk I will describe some aspects of this categorical correspondence. I hope to indicate the relationship to existing ideas in the subject: particularly to Taylor--Wiles--Kisin patching, but also to the work of Colmez and Paskunas, and to recent work of Johansson--Newton--Wang-Erickson. But more than this, I hope to indicate some of the underlying philosophy of the correspondence: what it means to represent the category of representations of a group geometrically, and why stacks (rather than just varieties) play a key role.