The main object of study in algebraic geometry is a variety, which is locally the solution set to polynomial equations. One fundamental research direction is the classification of these objects. In this talk, I'll introduce the idea of a moduli (or parameter) space for algebraic varieties. There will be many examples!

Historically, topological K-theory and its Bott periodicity have been very useful in solving key problems in algebraic and geometric topology. In this talk, we will explore the periodicities of higher real K-theories and their roles in several contexts, including Hill--Hopkins--Ravenel’s solution of the Kervaire invariant one problem. We will prove periodicity theorems for higher real K-theories at the prime 2 and show how these results feed into equivariant computations. We will then use these periodicities to measure the complexity of the RO(G)-graded homotopy groups of Lubin--Tate theories and to compute their equivariant slice spectral sequences. This is joint work with Zhipeng Duan, Mike Hill, Guchuan Li, Yutao Liu, Guozhen Wang, and Zhouli Xu.

Robust cellular function emerges from inherently stochastic components. Understanding this apparent paradox requires innovations in connecting mechanistic models of molecular-scale randomness with statistical approaches capable of extracting structure from large-scale, heterogeneous datasets. This talk presents a framework for inferring subcellular gene expression dynamics from static spatial snapshots of mRNA molecules obtained from single-molecule imaging. By linking spatial point processes with tractable solutions to stochastic PDEs, we recover dynamic parameters efficiently and without large-scale simulation. I’ll highlight recent theoretical results, including how cell-to-cell heterogeneity improves inference, and discuss extensions to transcriptional bursting, feedback, and cell-cycle effects. The work illustrates how combining mechanistic modeling with modern machine learning can propel new insights into complex biological systems.

Markov Decision Processes are the major model of controlled stochastic processes in discrete time. Value iteration (VI) is one of the major methods for finding optimal policies. For each discount factor, starting from a finite number of iterations, which is called the turnpike integer, value iteration algorithms always generate decision rules which are deterministic optimal policies for the infinite-horizon problems. This fact justifies the rolling horizon approach for computing infinite-horizon optimal policies by conducting a finite number of value iterations. In this talk, we will first discuss the complexity of using VI to approximately solve MDPs, and then introduce properties of turnpike integers and provide their upper bounds.