It is well-known that if a tensor category has an abelian algebra object A, one obtains a new category, essentially by tensoring over A. An important class of such algebra objects come from conformal inclusions for loop groups. While these algebra objects have been known for a long time, an explicit description of the corresponding categories was only recently found.

They are somewhat surprisingly closely related to representation categories of the isomeric quantum Lie super algebras. This talk is based on joint work with Edie-Michell and a paper by Edie-Michell and Snyder.

Recently, members of our group proved impressive results on the reduced C*-algebras of free groups—and, more generally, hyperbolic groups. Following the same general strategy, but using quite different methods, I obtain analogous results for higher-rank lattices (e.g., cocompact discrete subgroups of SL3(ℝ)). In the talk I’ll survey the structural properties of interest and outline the main ideas of the proofs.

Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times 10^{108}$. So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. In this work we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector itself is too large to store.

AI is taking off and we could say we are living in “the AI Era”.  Progress in AI today is based on mathematics and statistics under the covers of machine learning models.  This talk will explain recent work on AI4Crypto, where we train AI models to attack Post Quantum Cryptography (PQC) schemes based on lattices. I will use this work as a case study in training ML models to solve hard math problems in practice.  Our AI4Crypto project has developed AI models capable of recovering secrets in post-quantum cryptosystems (PQC).  The standardized PQC systems were designed to be secure against a quantum computer, but are not necessarily safe against advanced AI!  

Understanding the concrete security of these standardized PQC schemes is important for the future of e-commerce and internet security.  So instead of saying that we are living in a “Post-Quantum” era, we should say that we are living in a “Post-AI” era!

The Lévy-Khintchine theorem is a classical result in Diophantine approximation that describes the growth rate of denominators of convergents in the continued fraction expansion of a typical real number. We make this theorem effective by establishing a quantitative rate of convergence. More recently, Cheung and Chevallier (Annales scientifiques de l'ENS, 2024) established a higher-dimensional analogue of the Lévy-Khintchine theorem in the setting of simultaneous Diophantine approximation, providing a limiting distribution for the denominators of best approximations. We also make their result effective by proving a convergence rate, and in addition, we establish a central limit theorem in this context. Our approach is entirely different and relies on techniques from homogeneous dynamics.

Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

1) How to characterize the boundary trace Dirichlet space in a concrete way?

2) How does the boundary trace process behave? 

Based on a joint work with Shiping Cao.

The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a Euclidean space — such submanifolds are said to have cylindrical tangent cones at these singularities. Despite work in many contexts on minimal submanifolds with such singularities, the only known explicit examples at present are global products or involve extra structure (e.g. Kahler subvarieties). In this talk, I will describe a method for constructing infinite-dimensional families of non-product minimal submanifolds in arbitrary codimension whose singular set is itself an analytic submanifold. The construction uses techniques from the analysis of singular elliptic operators and Nash-Moser theory. This talk is based on joint work with Rafe Mazzeo.

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new close connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and number theory. In this talk, we will survey a few of these stories, and discuss some recent developments. Based on joint works with Alex Cohen and Dmitrii Zakharov. 

Deep neural networks have been widely adopted for climate prediction tasks and have achieved high prediction accuracy across many problems. However, their decision-making processes remain opaque, and the complexity of these models poses significant challenges for interpretation. A recent theoretical breakthrough, "Recursive Feature Machine" (RFM), provides an alternative methodology for climate prediction that is interpretable and data efficient. Applying RFM to El Niño–Southern Oscillation (ENSO) prediction yields promising interpretability results and offers insights into the most influential geographical features that the model learns from training data. The method is clean, easy to implement, and can be generalized to a broad range of scientific fields.

How in the world did they get those crazy pictures of electron orbitals? Those chemists had to have talked to somebody about it! It turns out they talked to math people (probably physicists, but physicists talk to math people, and so on). These orbitals can actually be derived in not-too-bad a way using representation theory. We'll go over what electron orbitals are, how they show up in the periodic table, how representation theory gets involved, and how to derive the electron orbitals ourselves. We will even find orbitals that are bigger than the highest electron on Oganesson! We'll hopefully also understand what physicists and engineers mean when they say they have a "tensor." I've also been studying the periodic table using mnemonic devices lately, so you'll be sure to hear about that.

Elliptic surfaces are complex surfaces with two discrete invariants, $g$ and $d>0$. We will discuss the moduli and Hodge theory of these surfaces for small values of $(g,d)$. The case $(g,d)=(1,1)$ is particularly interesting, in view of a new conjectural Fourier-Mukai type correspondence. It also provides a test case of the Hodge Conjecture in dimension 4.