This is the website for the Algebraic and Enumerative Combinatorics Seminar at the University of Waterloo. We view algebraic combinatorics broadly, explictly including algebraic enumeration and related asymptotic and bijective combinatorics as well as algebraic combinatorics as it appears in pure algebra and in applications outside mathematics.
We begin with a pre-seminar which is designed to get participants up to speed on useful and interesting background for the talk. It will be at the level for beginning grad students. Then there will be a short coffee break followed by the seminar itself.
Our audience consists principally of combinatorics faculty and grad students. Seminar talks are 50 minutes with questions following.
If you are speaking, we need your abstract at least a week in advance so it can make the deadline for the Friday math faculty seminar mailing.
Fall 2025
Usual location and time: 1:30 pre-seminar, 2:30 seminar, both in MC 6029.
Let $u$ be a word over the positive integers. Motivated in part by a question of representation theory, we study the centralizer set $C(u)$, which consists of words $w$ for which $wu$ and $uw$ are Knuth-equivalent. In particular, we characterize $C(u)$ when $u$ has few letters or is a certain type of decreasing sequence, and we address related enumerative questions.
September 18: There will be a watch party for a relevant talk from this ICERM workshop.
September 25: Jesse Huang, A Combinatorial Gateway to Calabi-Yau Toric Geometry Click here for abstract
The goal of this talk is to introduce Calabi-Yau toric geometry from a purely combinatorial perspective, through the rich structures carried by an embedded bipartite graph on a torus called a dimer model.
In the pre-talk, we will demonstrate how matchings, tilings, and quivers naturally encodes deep geometric and physical content. No background in algebraic geometry will be assumed; instead, we’ll build the story from the ground up, with an emphasis on visual intuition and discrete structures.
Next, I will discuss how dimer models simultaneously encode the data of a toric Calabi-Yau singularity and its mirror dual, unifying perspectives from the B-model and A-model of homological mirror symmetry through noncommutative algebras. We will proceed with some recent developments and open problems, including connections to the symplectic geometry of Landau-Ginzburg models and Van den Bergh’s noncommutative resolution conjecture for toric Gorenstein affine singularities. To conclude the talk, we will discuss an example of a higher dimensional generalization and its connection to my recent research works.
This talk is also partially based on two undergraduate research projects in the present semester, where we study variations of dimer models of the same lattice polygon from different choices in the fast inverse algorithm, and implications on the two sides of mirror symmetry: - (DRP) Dimer Variation and Geometry of Landau-Ginzburg Models, with Kenneth Xiao and Elizabeth Cai (A-model) - (NSERC USRA) Dimer Variation and NCCR Mutations, with Filip Mildrag and Elana Kalashnikov (B-model)
October 2: Nathan Pagliaroli, Enumerating planar stuffed maps as hypertrees of mobiles Click here for abstract
A planar stuffed map is an embedding of a graph into the 2-sphere, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to the 2-sphere with multiple boundaries. This is a generalization of planar maps whose complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to a disc. In this talk I will outline my work in constructing a bijection between bipartite planar stuffed maps and collections of integer-labelled trees connected by hyperedges such that they form a hypertree. This bijection directly generalizes the Bouttier-Di Franceso-Guitter bijection between bipartite planar maps and mobiles. Additionally, we show that the generating functions of these trees of mobiles satisfy both an algebraic equation, generalizing the case of ordinary planar maps, and a new functional equation. As an example, we explicitly enumerate a class of stuffed quadrangulations.
October 9: Graham Denham, Distances in trees and inequalities for matroids Click here for abstract
The distance matrix of a tree appears in a range of contexts, from phylogenetics to physical chemistry. I will describe a new result about the spectrum of this matrix, one that gives affirmative answers to two questions about matroid positivity properties. These are both strengthenings of Mason's conjectures about the log-concavity of sequence of numbers of independent sets of a matroid, proposed by Igor Pak and by Giansiracusa, Rincón, Schleis, Ulirsch, respectively.
This is joint work with Federico Ardila, Sergio Cristancho, Chris Eur, June Huh, and Botong Wang.
October 16: No seminar on account of reading week
October 23: Krystal Guo, Counting Substructures in Hypergraphs with Spectrum Click here for abstract
Jacobi’s classical result expresses the generating function for closed walks at a vertex of a graph as the ratio of two characteristic polynomials. We find a hypergraph analogue of this relationship, showing that the same rational function for a hypergraph also counts combinatorial substructures within it, called infragraphs. We use Viennot’s Heaps of Pieces framework to establish this result for the adjacency tensor of a hypergraph. As an immediate consequence, we obtain an alternative proof for the monotonicity of the principal eigenvalue of a hypergraph. This is based on joint work with Joshua Cooper and Utku Okur (arXiv: 2411.03567).
The Martin invariant of an even degree regular graph is an integer, defined by a recurrence at a vertex. In this talk, we define this invariant and review some of its properties, and then focus on the following question: which graphs minimise the Martin invariant?
November 6: Leigh Foster, Tilings of Benzels (and other finite regions) in the hexagon grid Click here for abstract
In 1990, Conway and Lagarias introduced tilability criteria for tilability for finite regions of the hexagon and square grids. In the same year, Thurston expanded upon their work, introducing the height function criterion. We will discuss some new results in tilability: A new tilability criteria via the $\textsf{SL}_2(\mathbb{C})$ double dimer model, and enumeration of tilings of special regions called Benzels, introduced in 2020 by Propp, using a technique called compression. If time allows, we will also discuss ongoing work that expands Thurston's height function to stone-and-bone tilings of the hexagon grid.
November 13: Pierre Popoli
November 20: No seminar on account of Tim Miller’s defense
The set of orthonormal bases for $k$-planes in $\mathbb{R}^n$ is cut out by the equations $X*X^T = I$ where $X$ is a $k \times n$ matrix of variables and $I$ is $k \times k$ identity. This space, known as the Stiefel manifold $\textrm{St}(k,n)$, generalizes the orthogonal group and can be realized as the homogeneous space $O(n)/O(n-k)$. Its algebraic closure gives a complex affine variety, and thus, it has a degree.
I will discuss our derivation of these degrees. Extending 2017 work on the degrees of special orthogonal groups, joint work with Fulvio Gesmundo gives a combinatorial formula in terms of non-intersecting lattice paths. This result relies on representation theory, commutative algebra, Ehrhart theory, polyhedral geometry, and enumerative combinatorics.
I will conclude with some open problems inspired by these objects.