In this talk, we discuss a method using Zariski localization to study how singularities of certain algebras such as Rees algebras or rational localizations behave under perturbation of defining ideals. If time permits, I will talk about a potential application to the almost purity theorem. 

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.

Weingarten calculus provides a representation-theoretic framework for evaluating Haar integrals of products of matrix entries over compact groups. In this talk, I will present a polynomial method for $U_N$ and $SU_N$ Weingarten calculus, focusing on contingency tables, Kostka operators, and polarization. Generating polynomials encode matrix-entry integrals, while contingency tables organize commutative monomials and lead to monomial integral formulas through Kostka-type operators. Polarization then restores the ordered information needed to recover link integrals from the same generating polynomial. I will also explain how the $SU_N$ theory differs from the $U_N$ theory: the determinant-one condition introduces shifted matching conditions and determinant powers in the character expansion. Finally, I will discuss an application to the Alon--Tarsi conjecture, where the special determinant type $SU_N$ integral recovers the signed difference between even and odd Latin squares and leads to new combinatorial interpretations through rectangular symmetric-group characters and permutation factorizations.

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]

Non-linear Random Matrix Theory (RMT) has recently emerged as a powerful paradigm for the theoretical understanding of deep learning theory. Throughout recent works, a universality principle, the \textit{Gaussian Equivalence Theorem} (GET), has become an indispensable tool allowing for the behavior of complex nonlinear neural networks to be understood through tractable linear kernel models. This thesis contributes to this emerging field, first by using the GET universality principle to derive a novel scaling law in Neural Tangent Kernel (NTK) regression, and second by studying the implications of this idealized linear equivalence on a high-dimensional nonlinearly separable dataset.