We determine the $\tau^n$-torsion in the first 5 lines of the $E_2$ page of the $\mathbb{C}$-motivic Adams spectral sequence using the techniques of Burklund-Xu. In particular, every element in this range is either $\tau^1$-torsion or $\tau$-free. We also show that $\tau^n$-torsion elements can appear only in Adams filtration at least $2n+2$ and give further evidence of a possible $3n$ bound.

In Part I, we provide an explicit version of the Eichler--Selberg trace formula for Shimura curves with level structure over the rationals. As an application, we provide an algorithm to compute the isogeny decomposition of the Jacobian of Shimura curves into modular abelian varieties using the method that Rouse--Sutherland--Zureick-Brown developed for classical modular curves. We also give a trace formula for definite quaternionic modular forms over the rationals.

In Part II, we compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of $\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.

We also report progress on compiling a corresponding list in genus 7 over $\mathbb{F}_2$ (for which explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_7$ are also available) and genus 5 over $\mathbb{F}_3$, where the censuses are complete in all except for the generic strata in both cases.

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.

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.