I will give an introduction to Brauer groups as they appear in a few different contexts: fields and algebraic varieties, topological spaces, and complex manifolds. In each context, one meets the same two problems, known as Grothendieck's question and the period-index problem. I will explain what is known and what is open, and describe some recent progress in all three cases.

A compact hyper-Kähler manifold is an extremely special compact Kähler manifold, whose local charts are given by the quaternionic space H^n. It naturally appears in certain classification theorems in both differential and algebraic geometry. Its rigid and special nature is well reflected in its (singular) cohomology. I will introduce the definition of a compact hyper-Kähler manifold and its basic cohomology theory. We will focus on its Hodge theoretic behavior.

Given the two-sphere, if we consider the standard rotation with an irrational angle, we see that the north pole and the south pole are the only periodic points, aka fixed points of all its iterates. A celebrated theorem of Franks says that, for any area-preserving diffeomorphism of the two-sphere, if it has at least three fixed points, then it has infinitely many periodic points. In this lecture, I will discuss a recent generalization of Franks' theorem to all toric symplectic manifolds: if a Hamiltonian diffeomorphism thereon has at least the toric fixed points worth of fixed points plus one, then it has infinitely many periodic points. Surprisingly, quasi-maps in certain disguised forms play an important role in the proof.

Theta correspondence is a classical and very useful theory to lift automorphic representation. Under certain circumstances, there is a nice criterion to determine whether a theta lifting is non-trivial in terms of local-global compatibility and special values of L-function. When the theta lifting is trivial due to local reason, people expect there exists certain arithmetic theta lifting and a criterion of triviality in terms of derivatives of L-function. In this talk, I will survey this theory and explain how this fits into the broader Kudla program.

Discrete data sets (e.g., graphs or points clouds) often turn out to have manifold structure. In the field of geometric data analysis (GDA), we use ideas from differential geometry and topology to analyze such data. In this talk, we’ll focus on the role of discrete curvature in data science. First we’ll introduce the idea of “Ollivier-Ricci curvature”, a definition of graph curvature that is inspired by ideas in optimal transport. Then we’ll discuss ways that curvature is connected to various other techniques in GDA, such as manifold learning and topological data analysis. I’ll close by presenting some of my ongoing research and open questions related to discrete curvature.

In this talk I will motivate the new theory of analytic geometry constructed by Clausen and Scholze via condensed mathematics. We will see different examples of classical analytic spaces and how this theory allows a clean framework to study all of them simultaneously.

Geometric representation theory studies the interactions between representation theory and algebraic/symplectic geometry. In this talk, I will give a friendly introduction to the concept of a Fukaya category associated to a symplectic manifold and illustrate its role in geometric representation theory with two elementary examples: quiver 3-folds and hypertoric varieties.

The last 20 years has seen remarkable growth in studying a class of random growth models said to lie in the KPZ universality class. This class consists of (1+1)-dimensional models (1 time and 1 space dimension) that describe physical phenomena such as the growth of forest fires, bacteria colonies, tumors, and many others. In this field, we see deep connections to many areas of mathematics, including representation theory, combinatorics, and PDEs. I will give an overview of KPZ universality and discuss how my research fits in to this bigger picture. Topics of my own work may include joint works with Ofer Busani, Sean Groathouse, Firas Rassoul-Agha, and Timo Seppalainen.