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!

Diffusion-based generative models have become the standard for image generation. ODE-based samplers and flow matching models improve efficiency, in comparison to diffusion models, by reducing sampling steps through learned vector fields. However, the theoretical foundations of flow matching models remain limited, particularly regarding the convergence of individual sample trajectories at terminal time—a critical property that impacts sample quality and being critical assumption for models like the consistency model.

In this paper, we advance the theory of flow matching models through a comprehensive analysis of sample trajectories, centered on the denoiser that drives ODE dynamics. We establish the existence, uniqueness, and convergence of ODE trajectories at terminal time, ensuring stable sampling outcomes under minimal assumptions. Our analysis reveals how trajectories evolve from capturing global data features to local structures, providing the geometric characterization of per-sample behavior in flow matching models. We also explain the memorization phenomenon in diffusion-based training through our terminal time analysis. These findings bridge critical gaps in understanding flow matching models, with practical implications for sampling stability and model design. This is a joint work with Zhengchao Wan, Gal Mishne and Yusu Wang.

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.

The presentation will cover new results on modal clustering. We first provide a unifying view of this topic, which includes two important non-parametric approaches to clustering that emerged in the 1970s: clustering by level sets or cluster tree as proposed by Hartigan; and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. We will draw a close connection between these two views by 1) showing that the gradient flow provides a way to move along the cluster tree; and 2)  by proposing two ways of obtaining a partition from the cluster tree—each one of them very natural in its own right—and showing that both of them reduce to the partition given by the gradient flow under standard assumptions on the sampling density. We will then establish some consistency results for various methods that have been proposed for modal clustering, including the famous Mean Shift algorithm proposed by Fukunaga and Hostetler in that same article. If time permits, we will conclude by a broader discussion of what is meant by clustering in Statistics, and suggest a set of axioms for hierarchical clustering that lead to Hartigan's definition. 

Joint work with Wanli Qiao (George Mason University) and Lizzy Coda (UC San Diego).

Chern classes are an important object in several areas of mathematics. At first glance, definitions of Chern classes across areas of mathematics may not appear similar. In this talk, we will see different definitions of Chern classes and will discuss the equivalency of these definitions using an axiomatic approach.