Approximating data sets by graphs and simplicial complexes has been shown to be a very useful way to obtain qualitative information about data, and more recently has been shown to similarly contribute to artificial intelligence.  I will discuss the mathematics around this, with examples from various domains.  

BIO: Gunnar Carlsson is the Ann and Bill Swindells Professor Professor of Mathematics, Emeritus, at Stanford University, and a pioneer in the field of computational topology. His research focuses on the application of topological methods  to the analysis of high-dimensional, complex data, a discipline known as Topological Data Analysis (TDA). Professor Carlsson is perhaps best known for leading the "Topological Methods in Data Analysis" project (supported by DARPA), which catalyzed the development of persistent homology and mapper algorithms. Beyond his academic contributions, he co-founded Ayasdi, a company dedicated to utilizing TDA for industrial-scale machine learning and data science. He holds a Ph.D. from Stanford and has previously held faculty positions at the University of Chicago, the University of California, San Diego, and Princeton University. 

Symplectic resolutions arise in representation theory (Springer resolution), algebraic geometry (Hilbert--Chow morphism), and mathematical physics (3D mirror symmetry). There is a program to classify all possible symplectic resolutions of a given singularity. This classification simplifies when the singularity is conical, as it suffices to resolve any neighborhood of the cone point.

In ongoing work with Travis Schedler, we extend the perspective beyond conical singularities. Surprisingly, local resolutions of conical neighborhoods extend and glue uniquely to a global resolution, provided they are monodromy-free and chosen compatibly.

Anomaly detection is widely used in real-world applications such as industrial fault detection, quality control, cybersecurity, fraud detection, healthcare monitoring, and scientific data analysis. In these scenarios, abnormal patterns are often rare but critical. However, practical anomaly detection is challenging because real-world data are usually noisy, incomplete, high-dimensional, graph-structured, or collected from heterogeneous domains, while reliable anomaly labels are often limited or unavailable.

This talk presents a line of research on robust learning for anomaly detection in complex and imperfect data. I will discuss methods for tabular anomaly detection under noise and missing values, graph-level anomaly detection, automatic hyper-parameter optimization, semi-supervised anomaly detection, and universal outlier detection across diverse domains. Together, these works aim to develop anomaly detection methods that are robust, adaptive, and generalizable for real-world applications.

This talk presents a polynomial method for studying Haar integrals over $U_N$ and $SU_N$, motivated by matrix-model partition functions and link integrals in lattice gauge theory. Instead of evaluating ordered matrix-entry integrals through invariant tensor bases or entrywise differentiation, we package them into generating polynomials and organize their coefficients using polarization, commutative monomials, and contingency tables. This leads to the Kostka-operator formula for $SU_N$ monomial integrals. As an application, we relate a one-sided $SU_N$ integral to the coefficient of the full-support monomial in $(\det X)^N$, which equals the difference between the numbers of even and odd Latin squares, giving a new combinatorial perspective on the Alon--Tarsi conjecture.

This dissertation develops a data-driven framework for equilibrium discovery in parameterized dynamical systems. The central idea is to represent the solution set through a parameter-solution neural network (PSNN), which learns a scalar landscape Φ(u,θ) on the product space of state variables and parameters. Peaks of this learned landscape encode steady states, allowing the method to recover varying numbers of equilibria across parameter regimes. The dissertation further develops adaptive refinement procedures and classifier-assisted algorithms to improve the localization of closely spaced solutions, infer stability, and remain robust under incomplete observations. Finally, it establishes approximation-theoretic guarantees for the PSNN architecture and demonstrates the framework on benchmark dynamical and biochemical reaction systems.

A variety is unirational if it admits a dominant rational map from projective space. In characteristic zero, global tensor forms obstruct unirationality. This is the principle behind the Harris–Mumford theorem (1982): M_g is of general type, and a fortiori not unirational, for g large. In positive characteristic the picture is far wilder, owing to the existence of inseparable maps, and as a result the unirationality of only a handful of moduli spaces is understood.

I will introduce new techniques for obstructing unirationality in positive characteristic, inspired by methods for proving hyperbolicity in complex geometry. As applications, I give a counterexample to Shioda's 1977 conjecture that a simply connected surface in positive characteristic is unirational if and only if it is supersingular. I also show that many Hilbert modular varieties in positive characteristic are not unirational or even covered by rational or elliptic curves.

[pre-talk at 3:00PM]

In this talk, I will begin by introducing graphons, which are infinite-size limits of adjacency matrices of sequences of growing graphs. I will then define graph reaction-diffusion (RD) equations, which are systems of differential equations that are defined on the nodes of a graph. For a sequence of growing graphs that converges to a graphon, the solutions of the sequence of graph RD equations also converge. The limiting solution solves a nonlocal differential equation that we call a graphon RD equation. Furthermore, the graph RD equation is related to a stochastic birth-death process on graphs. I will show that this birth-death process converges to the graphon RD equation via a hydrodynamic limit.

In 1958, Hirzebruch produced a generating function for the Hodge numbers of a smooth hypersurface Z in P^n, and in 1969, Griffiths produced the residue map from the space of polynomials to differential forms. If a group G acts linearly on Z, the Griffiths residue map is G-equivariant. This map allows us to describe the primitive cohomology of Z in terms of graded pieces of a particular ring — the Griffiths ring. In this talk we generalize Griffiths' construction to smooth hypersurfaces in Grassmannians Gr(k,n), assuming some mild divisibility constraints on k,n, and the degree of the hypersurface.