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Organizing Committee
University of York
University of Connecticut
Dartmouth College
University of California, Davis
Worcester Polytechnic Institute
Abstract
Can machines prove theorems? Can they have mathematical ideas? On one hand, category theory offers a formalism for axiomatising ideas from machine learning. On the other hand, mathematicians are excited about the prospect of utilising machine learning techniques to spot new patterns in vast swathes of combinatorial data and hence formulate new conjectures.
The purpose of this workshop is to bring together experts from across algebraic combinatorics, category theory, and machine learning in order to make headway on topics at the intersection of these fields.
Application Deadline: May 31, 2025
The purpose of this workshop is to bring together experts from across algebraic combinatorics, category theory, and machine learning in order to make headway on topics at the intersection of these fields.
Application Deadline: May 31, 2025
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Workshop Schedule
Monday, September 15, 2025
8:30 - 9:00 AM EDT
Check In
11th Floor Collaborative Space
9:00 - 9:10 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:10 - 10:00 AM EDT
Plethysm, symmetric chain decompositions and Erhart polynomials
11th Floor Lecture Hall
Speaker
Mike Zabrocki, York University
Session Chair
Anne Schilling, University of California, Davis
Abstract
I will present a long standing open problem of finding a symmetric chain decomposition of the poset of partitions in a w x h rectangle. I will show how this problem is related to that of finding a combinatorial interpretation for the plethysm coefficients of an SL_2(C) representation and show that certain generating functions of these plethysm coefficients are rational. Finally, I will relate this problem to counting integer points in a polytope.
This is joing work with Álvaro Gutiérrez, Rosa Orellana, Franco Saliola and Anne Schilling
10:15 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:50 AM EDT
Cluster deep loci and cluster algebras over finite fields
11th Floor Lecture Hall
Speaker
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Session Chair
Anne Schilling, University of California, Davis
Abstract
The deep locus of a cluster algebra A consists of the complement to all the cluster tori inside Spec(A). It is a challenging problem to determine when it is nonempty, partly because there are usually an infinite number of cluster tori. By passing to finite fields, one can hope to obtain a finite number of cluster tori covering the complement of the deep locus, and lift these to characteristic zero. While this turns out not to be the case, studying cluster tori over finite fields leads to interesting combinatorics. In particular, we will see how to recover from a cluster-theoretic perspective the "hexagonal edges" of the universal polytope of all partitions of an n-gon. This is based in joint works with different subsets of {Marco Castronovo, Mikhail Gorsky, Daniel Pérez Melesio, David Speyer}, and I won't assume prior knowledge of cluster algebras.
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:50 PM EDT
Triangular (q,t)-Schroder polynomials and Khovanov-Rozansky homology of Coxeter knots
11th Floor Lecture Hall
Speaker
Nicolle Gonzalez, UC Berkeley
Session Chair
Chris Bowman, University of York
Abstract
Catalan numbers are among the most ubiquitous objects in mathematics, arising naturally in combinatorics, representation theory, geometry, and many other areas. Although there are various polynomial generalizations of these numbers, particularly fruitful are the so-called (q,t)-Catalan polynomials. Among many other things, these polynomials provide a direct link between combinatorial objects, such as Dyck paths and parking functions, and the Khovanov-Rozansky homology of so-called torus knots. In this talk, I will present the new family of triangular Schroder polynomials which contain the (q,t)-Catalans as a special case and compute the hook components of the Shuffle theorem under any line. In particular, these polynomials coincide with the Poincare polynomials of the Khovanov-Rozansky homology of Coxeter knots. As a consequence we prove a conjecture of Oblomkov-Rasmussen-Shende for a certain family of cabled torus knots and show that the triangular Schroder polynomials agree with the EHA superpolynomial of Galashin and Lam in this case.
3:05 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:20 PM EDT
The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale
11th Floor Lecture Hall
Speaker
Terence Tao, UCLA
Session Chair
Chris Bowman, University of York
Abstract
I will report on the Equational Theories Project, which successfully determined the logical relationships between the 4694 simplest equational laws of a single binary operation, by coordinating contributions from a diverse set of both human and computer-generated sources, all formalized within a common Lean framework.
4:30 - 6:00 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, September 16, 2025
9:10 - 10:00 AM EDT
Abelian categorifications
11th Floor Lecture Hall
Speaker
Jonathan Brundan, University of Oregon
Session Chair
Rosa Orellana, Dartmouth College
Abstract
It is sometimes useful to systematically study Abelian categories equipped with some additional structure, e.g., tensor categories. I will talk about some less well known families of examples, Abelian categorifications, which appear in the representation theory of classical groups and related algebras. The simplest situation is that of a Heisenberg categorification, which is an Abelian category equipped with extra structure making it into a module category over a (monoidal) Heisenberg category. Heisenberg categorifications are ubiquitous in many parts of type A representation theory, and a remarkably rich structure theory has emerged which involves also the parallel notion of a Kac-Moody categorification. This has its origins in various works of Chuang, Khovanov, Lauda and Rouquier. It is the most highly developed GL branch of a more general story. There are also emerging OSp, P and Q branches associated to the other three classical families of supergroups over algebraically closed fields.
10:15 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:50 AM EDT
Computational Complexity in Algebraic Combinatorics: recent developments
11th Floor Lecture Hall
Speaker
Greta Panova, University of Southern California
Session Chair
Rosa Orellana, Dartmouth College
Abstract
Representation theoretic multiplicities (e.g. Kronecker, plethysm coefficients) are subject to major open problems in algebraic combinatorics. At the same time these and related quantities appear in Geometric Complexity Theory in the search for multiplicity obstructions which separate computational complexity classes like VP vs VNP. Most recently they have also been considered in quantum computing. We will discuss these connections and recent advances on these topics.
12:00 - 12:05 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:50 PM EDT
Left regular band semigroups with symmetry
11th Floor Lecture Hall
Speaker
Patricia Commins, University of Minnesota
Session Chair
Kyu-Hwan Lee, University of Connecticut
Abstract
Left regular bands (or LRBs) are a special family of finite, noncommutative semigroups which appear surprisingly frequently in algebraic combinatorics and discrete geometry. Originally studied for their connections to Markov chains, LRBs have more recently been studied for the rich yet tractable representation theory of their semigroup algebras. Many of the LRBs in the literature come equipped with natural symmetry groups which act by semigroup automorphisms. In such cases, one can study the structure of the invariant subalgebra of the semigroup algebra. In this talk, we will discuss joint work with Benjamin Steinberg which approaches understanding the representation theory of these invariant subalgebras through group-equivariant poset topology.
3:05 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:20 PM EDT
Machine learning and example-making in combinatorics
11th Floor Lecture Hall
Speaker
Jordan Ellenberg, University of Wisconsin
Session Chair
Kyu-Hwan Lee, University of Connecticut
Abstract
I will report on some experimental results with various co-authors in industry and academia centered on the question of: how can we coax machine learning models to generate examples of interest in combinatorics? To what extent can we generate material that's useful for a general class of problems, not just one instances? What characteristics of a problem make it amenable to these methods at present?
Wednesday, September 17, 2025
9:10 - 10:00 AM EDT
Some questions around the Bruhat order
11th Floor Lecture Hall
Speaker
Nicolas Libedinsky, Universidad de Chile
Session Chair
Mike Zabrocki, York University
Abstract
The Bruhat order hides a lot of combinatorial structure, yet two of its most natural questions remain wide open: Given an interval, how many elements does it contain? And how can we classify these intervals as posets? Recent results suggest surprisingly elegant answers for these questions, and for affine Weyl groups, we have found that a small part of the poset almost distinguishes between non-isomorphic Bruhat intervals. Perhaps ideas from machine learning could be the key to closing this gap.
10:15 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:50 AM EDT
Monoidal categorification and crystals
11th Floor Lecture Hall
Speaker
Euiyong Park, University of Seoul
Session Chair
Mike Zabrocki, York University
Abstract
Monoidal categorification is a categorical-approach to studying (quantum) cluster algebras from the perspective of monoidal categories. In this talk, I will discuss the categorification of quantum groups using quiver Hecke algebras by Khovanov-Lauda and Rouquier, and explain how quiver Hecke algebras provide a monoidal categorification following the work of Kang-Kashiwara-Kim-Oh. I will then explain how the crystal theory can be used in studying the monoidal categorification, and present my recent result on quantum twist automorphisms and crystals. This is partially based on joint work with Woo-Seok Jung (arXiv:2507.01306).
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:50 PM EDT
Fibers of Maps to Totally Nonnegative Spaces
11th Floor Lecture Hall
Speaker
Patricia Hersh, University of Oregon
Session Chair
Zajj Daugherty, Reed College
Abstract
Fomin-Shapiro, Lusztig, and several others have studied maps to totally nonnegative spaces whose fibers encode the nonnegative real relations amongst exponentiated Chevalley generators. We prove that the stratification on each such fiber induced by the natural stratification of $\RR_{\ge 0}^d$ is a cell decomposition, doing so by providing a parametrization for each stratum. We also show that the face poset for this cell decomposition is the face poset of a regular CW complex, namely of the interior dual block complex of a subword complex. This talk will focus on examples illustrating some of the main ideas that go into this work as well as background, motivations, and some further open questions. This is joint work with Jim Davis and Ezra Miller.
3:05 - 4:00 PM EDT
4:00 - 4:50 PM EDT
Automated conjecturing and combinatorial geometry
11th Floor Lecture Hall
Speaker
Jillian Eddy, University of California, Davis
Session Chair
Zajj Daugherty, Reed College
Abstract
What makes a conjecture meaningful or interesting? We discuss a geometric perspective of automating the process of mathematical conjecturing. This theory is then applied via experiments in combinatorial geometry, particularly in understanding sequences derived from the facets of simple combinatorial 3-polytopes. We illustrate the potential of an agentic approach in machine-assisted mathematical research.
Thursday, September 18, 2025
9:10 - 10:00 AM EDT
Factorizations, centers, and the Jucys—Murphy elements of the Hecke algebra
11th Floor Lecture Hall
Speaker
Sarah Brauner, Brown University
Session Chair
Patricia Hersh, University of Oregon
Abstract
Given a permutation, there is a well-developed literature studying the number of ways one can factor it into a product of other permutations, subject to certain conditions. We initiate the analogous theory for the type A Iwahori Hecke algebra by generalizing the notion of factorization in terms of a family of elements called the Jucys—Murphy elements. A key ingredient in our analysis involves studying the character theory and center of the Hecke algebra, where many interesting questions and enigmas arise. One hope is that computational and machine learning tools can help elucidate these mysteries.
Joint work with Jose Bastidas, Mathieu Guay-Paquet, Alejandro Morales, GaYee Park and Franco Saliola.
10:15 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:50 AM EDT
Interpreting AI solutions to the word problem in automatic groups?
11th Floor Lecture Hall
Speaker
J. Elisenda Grigsby, Boston College
Session Chair
Patricia Hersh, University of Oregon
Abstract
There is a growing interest among both mathematicians and AI researchers to apply AI models to mathematical questions. Unfortunately, many AI models are black boxes. Experts understand poorly both the process by which they arrive at their predictions and how to interpret the predictions they give. In this largely expository talk, I will survey some recent work in mechanistic interpretability, an empirically-driven field whose goal is to extract algorithms from trained AI models. After reporting on findings from an undergraduate research project I co-directed (with C. Ashley) on mechanistic interpretation of several different AI model solutions to the word problem in Coxeter groups, I will suggest promising avenues for future research and comment on the challenges involved in pursuing them.
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:50 PM EDT
Surprises in Specht polynomial theory
11th Floor Lecture Hall
Speaker
Maria Gillespie, Colorado State University
Session Chair
Monica Vazirani, UC Davis
Abstract
Higher Specht polynomials were first defined by Ariki, Terasoma, and Yamada as a way to construct a basis of the coinvariant ring that respects the decomposition into irreducible S_n modules. They are higher degree versions of the ordinary Specht construction of the irreducible representations, obtained by applying a Young symmetrizer to an appropriate monomial. This construction has recently been generalized to form bases of several generalizations of the coinvariant ring.
We will discuss the general theory of higher Specht polynomials in this talk, focusing on the question: When does the resulting polynomial vanish? Some surprises show up in the code, and we ask whether machine learning tools can help address this problem as well. Some of this material will be on joint work with Raymond Chou.
3:05 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:20 PM EDT
TBA
11th Floor Lecture Hall
Speaker
Sergei Gukov, American Institute of Mathematics
Session Chair
Monica Vazirani, UC Davis
Friday, September 19, 2025
9:10 - 10:00 AM EDT
A probabilistic bijection for non-attacking fillings
11th Floor Lecture Hall
Speaker
Olya Mandelshtam, University of Waterloo
Session Chair
Joshua Swanson, University of Southern California
Abstract
Based on the connection to one-dimensional interacting particle models--the ASEP and the TAZRP--and the combinatorial objects (multiline queues) that make these connections explicit, a tableau formula was found for the modified Macdonald polynomial in terms of the so-called `queue inversion' statistic. This statistic naturally reflects the dynamics of the TAZRP. In this talk, we present a new tableau formula for the symmetric Macdonald polynomials, using the same queue inversion statistic on non-attacking tableaux. Central to our approach is a probabilistic bijection on non-attacking fillings that enables swapping of entries between columns. Our techniques extend to fillings of composition shapes, leading to symmetry results for permuted basement Macdonald polynomials.
10:15 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:50 AM EDT
ML and AI in Combinatorics Research: Some Examples
11th Floor Lecture Hall
Speaker
Blake Jackson, University of Connecticut
Session Chair
Joshua Swanson, University of Southern California
Abstract
Machine Learning and AI are beginning to change how we do combinatorics. In this talk, I’ll share how classification tools have helped uncover new insights into quiver mutation, and I’ll give a glimpse of an ongoing project that pushes these ideas further, with a look at what might be the next big leap: using AI not just to analyze combinatorics, but to create it—by automatically generating combinatorial bijections.
12:05 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:50 PM EDT
Characters of tilting modules, quotient by the Steinberg character, and Smith-Treumann localization
11th Floor Lecture Hall
Speaker
Pramod Achar, Louisiana State University
Session Chair
Christian Ikenmeyer, University of Warwick
Abstract
Let G be a connected reductive group over a field of positive characteristic. This talk is about a computational problem related "tilting modules" for G. In two recent papers, P. Sobaje proved some remarkable properties of expressions of the form "(character of a not-too-small tilting G-module)/(character of the Steinberg module)." In this talk, I will review Sobaje's results, and I will explain a geometric incarnation of these results in terms of parity sheaves and "Smith-Treumann localization" on the dual affine Grassmannian. If time permits, I will also discuss connections to some homological phenomena in the derived category of G-modules. This is joint work with P. Sobaje.
3:05 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:20 PM EDT
Superspace combinatorics
11th Floor Lecture Hall
Speaker
Brendon Rhoades, University of California, San Diego
Session Chair
Christian Ikenmeyer, University of Warwick
Abstract
The *superspace ring* is the bigraded algebra of differential forms on n-space. We study the quotient SR_n of superspace by the ideal generated by S_n-invariants with vanishing constant term. We give an explicit basis of SR_n and prove a 2018 conjecture of the Fields Institute Combinatorics Group on its bigraded S_n-module structure. Our methods use the theory of hyperplane arrangements. We present a combinatorial conjecture on a superspace quotient attached to any real arrangement. Joint with Robert Angarone, Patricia Commins, Trevor Karn, Satoshi Murai, and Andy Wilson.
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