Department of Mathematics Calendar

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Week of January 11, 2026

January 13, 2026
Special Guest Lecture	Positroids, knots, and q,t-Catalan numbers	Pavel Galashin -	4:00pm - KT 801
	Positroids, knots, and q,t-Catalan numbers	Pavel Galashin - Cornell University	4:30pm - TBA

January 14, 2026
Special Guest Lecture	Introduction to Spin Glass Theory	Yuxin Zhou - University of Chicago	10:00am - KT 801
Special Guest Lecture	Where can free waves concentrate	Ruixiang Zhang - institution (University of California, Berkeley)	11:15am - KT 801
Colloquium	The Convexity Conjecture, the Kahn-Kalai Conjecture, and introduction to k-thresholds	Jinyoung Park - NYU	4:00pm - KT 101

January 15, 2026
Special Guest Lecture	Mathematical Exploration and Discovery at Scale	Javier Gomez-Serrano - Brown University	10:00am - KT 801

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 101	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.