Abstracts

Week of January 11, 2026

January 13, 2026
Special Guest Lecture Positroids, knots, and q,t-Catalan numbers 4:00pm -
KT 801

Abstract: Open positroid varieties are certain subvarieties of the Grassmannian that arise in the study of total positivity and have surprising applications in many areas of mathematics and physics. After reviewing some history and background, I will discuss our recent joint work with Thomas Lam relating the cohomology of these varieties and their point counts over finite fields to knot invariants such as the HOMFLYPT polynomial and Khovanov–Rozansky homology. In particular, we show that the bigraded Poincaré polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. No background on the above objects will be assumed.
Positroids, knots, and q,t-Catalan numbers 4:30pm -
TBA

Abstract: Open positroid varieties are certain subvarieties of the Grassmannian that arise in the study of total positivity and have surprising applications in many areas of mathematics and physics. After reviewing some history and background, I will discuss our recent joint work with Thomas Lam relating the cohomology of these varieties and their point counts over finite fields to knot invariants such as the HOMFLYPT polynomial and Khovanov–Rozansky homology. In particular, we show that the bigraded Poincaré polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. No background on the above objects will be assumed.
January 14, 2026
Special Guest Lecture Introduction to Spin Glass Theory 10:00am -
KT 801

Abstract: Spin glasses are disordered magnetic systems that display remarkably complex collective behavior. Understanding these systems has challenged both physicists and mathematicians for decades. In the late 1970s, Giorgio Parisi proposed a revolutionary solution to the Sherrington–Kirkpatrick model. At the heart of his theory is the Parisi measure, an object that reveals the hidden organization of disorder. In this talk, I will survey what is now rigorously known about the Parisi measure and highlight the ​major open problems in the field.
Special Guest Lecture Where can free waves concentrate 11:15am -
KT 801

Abstract: Waves are ubiquitous in our daily life. Two best-known linear models are the free wave and free Schrödinger equations, whose simplest forms are very amenable to Fourier analysis. Still, a basic question—how large can a solution be, and where can it be large?—is surprisingly subtle and only partly understood, especially in higher dimensions. Over decades, it transpired that in order to answer this fundamental question, one often needs to understand whether and how much the solution can concentrate on important subsets of $\mathbb{R}^n$. I will discuss three kinds of such subsets (convex sets, semialgebraic sets and lattices) and their importance based on sample problems. Some of them have nice connections to nearby areas such as number theory, geometry and combinatorics.
Colloquium The Convexity Conjecture, the Kahn-Kalai Conjecture, and introduction to k-thresholds 4:00pm -
KT 205

Abstract: The “Convexity Conjecture” by Talagrand asks (very roughly) whether one can “create convexity” in constant steps regardless of the dimension of the ambient space. Talagrand also suggested a discrete version of the Convexity Conjecture and called it “my lifetime favorite problem,” offering $1,000 prize for its solution. We introduce a reformulation of the discrete Convexity Conjecture using the new notion of “k-thresholds,” which is an extension of the traditional notion of thresholds, introduced by Talagrand. Some ongoing work on understanding k-thresholds, along with a (vague) connection between the Kahn-Kalai Conjecture and the discrete Convexity Conjecture, will also be discussed. Joint work with Ascoli, He, and Talagrand.
January 15, 2026
Special Guest Lecture Mathematical Exploration and Discovery at Scale 10:00am -
KT 801

Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.
Quantum Topology and Field Theory Finiteness of Kauffman bracket skein modules 4:30pm -
KT 801

Skein modules are algebraic objects that somehow “encode” links in a 3-dimensional manifold, in a way reminiscent of homology; they have many interesting connections to physics, representation theory, knot theory (via the Jones polynomial) and non-commutative algebra. In this talk I will give a broad introduction and overview of the topic and then discuss a new, elementary proof (due to myself and Renaud Detcherry) of finite dimensionality of the Kauffman bracket skein modules, originally done by Gunningham-Jordan-Safronov.