Current students

Abstract: The mirror of a smooth Fano toric variety is a Laurent polynomial, and this Laurent polynomial encodes interesting enumerative data of the toric variety.  For example, the Jacobian ring of the Laurent polynomial is isomorphic to the small quantum cohomology ring of the toric variety. In this talk, I’ll describe this aspect of the (now basically classical) toric mirror theorem, and explain how this picture is expected to extend to other smooth Fano varieties. The Grassmannian’s strong combinatorial structure  makes it one of the nicest examples of non-toric Fano varieties to study in this context. I’ll present several mirror constructions for the Grassmannian that each reflect a different perspective on its geometry. Finally, I’ll discuss some extensions of these constructions to generalizations of the Grassmannian. 

Seminar: Algebraic and Enumerative Combinatorics Seminar
Date: Thursday, September 17, 2026
Time: 2:30 PM
Location: MC 5417
Speaker: Mike Zabrocki
Affiliation: York University
Title: Characters of the rook monoid
Abstract:

The partial permutations form a monoid known in different contexts as either the “symmetric inverse semigroup” or the “rook monoid.” In this talk I will define a family of symmetric functions that evaluate to the character values of the irreducible representations of the monoid. The basis also connects to the representation theory of the Schur-Weyl duality with the “propagating partition algebra.” Moreover, the structure coefficients of this basis interpolate between the Kronecker coefficients and the Littlewood-Richardson coefficients and we use it to develop some of the combinatorics of the connection.

This is joint work with Rosa Orellana and Alex Wilson.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

 

Acyclic List Colouring Locally Planar Graphs

Massimo Vicenzo | University of Waterloo

Abstract: A (vertex) colouring of graph is acyclic if it contains no bicoloured cycle. In 1979, Borodin proved that planar graphs are acyclically 5-colourable. In 2010, Kawarabayashi and Mohar proved that locally planar graphs are acyclically 7-colourable. In 2002, Borodin, Fon-Der-Flaass, Kostochka, Raspaud, and Sopena proved that planar graphs are acyclically 7-list-colourable. In this talk we discuss our result that locally planar graphs are acyclically 9-list-colourable—no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known.

This is joint work with Luke Postle and Evelyne Smith-Roberge. 

Title: A Proof of Nash-Williams’ Conjecture

Abstract: A central open question in extremal design theory is Nash-Williams’ Conjecture from 1970, namely that every triangle-divisible graph on n vertices (for n large enough) with minimum degree at least 0.75n has a triangle decomposition. In this talk, we discuss the history of the problem and our recent resolution of this conjecture, as well as other applications in design theory. We also overview the proof, highlighting the new techniques we developed to resolve the fractional version as well as the full conjecture. Joint work with Michelle Delcourt.

Abstract: This term we are running a learning seminar on homology and cohomology, following Chapters 2 and 3 of Hatcher.  In this first meeting, we will briefly talk about our plans for the seminar, and then will jump right into simplicial homology. All are welcome!

Abstract: 

In 2019, Google announced the demonstration of quantum supremacy by performing a computational task known as random circuit sampling on their 53-qubit quantum computer Sycamore. In their paper, they claimed that it would take classical computers 10000 years to perform the same task. Almost seven years has passed, and what happened to this claim since then? In this talk, I will provide a high-level update on the subsequent developments in quantum hardware experiments, classical simulation software, asymptotic classical simulation algorithms, and proving the hardness of classical simulation. This talk is aimed at a non-quantum computing audience, and no prior background in quantum computing is assumed.

Abstract: 

Given an LP with tall and skinny constraint matrix, we will exploit this property and study an algorithm invented by Clarkson [8]. Although this algorithm has
been around for over 30 years, there were no software or implementation that could be found online, nor there be any benchmarks for these special tall and skinny LP s. We will describe some variants and changes to the algorithm aiming for practical performancesto close this gap.

We also study a first order algorithm aimed for large scale LP s proposed by a group of researchers from Google [2], [3] called PDLP. And compare it with Clarkson’s algorithm.

Abstract: 

Second-order Interior Point Methods (IPM) have been studied extensively over the past 80 years, proving effective for conic optimization. They can produce high-precision approximate solutions in few iterations. Each iteration is computationally expensive: The core of each iteration is a large matrix inversion that scales poorly with the number of variables.

In large-scale applications, we cannot bear the per-iteration cost (perhaps due to lack of memory), so we instead turn to first-order methods. We study a first-order IPM that uses a low-rank update scheme to replace the matrix inversion with significantly lower per-iteration cost, and compare this to other first-order methods for solving LP at scale.

Abstract: Conic geometry encodes combinatorial properties of a convex program. Under a probabilistic model of the data, these combinatorial properties become random events. Their likelihood is the measure of a cone. We illustrate this view with a dual pair of questions. First, how much can a linear program be regularized before its solution changes? With random costs, the answer turns on the Gaussian measure of the solution's normal cone. Second, how many measurements are needed to separate a superposition of structured signals? Here, each signal's complexity is the statistical dimension of its descent cone. A convex program recovers the components once the measurement count exceeds the total complexity.

Abstract: 

A tournament $\T=(V,A)$ on $n$ vertices is an orientation of the complete graph $K_n$. The backedge graph of $T$ with respect to an ordering of $V$ is the undirected graph on vertex set $V$ whose edge set corresponds to the arcs directed from a later vertex to an earlier vertex in the ordering. Backedge graphs provide concise representations of the tournament. The algorithmic problem of determining whether a tournament admits a backedge graph in a given class of undirected graphs varies in complexity, and is often equivalent to computing parameters of tournaments, such as degreewidth when the backedge graph has bounded maximum degree \cite{Davot et al., 2023}. We extend the notion of degreewidth by introducing directional degreewidth, which separately bounds the left-degrees and right-degrees of vertices in addition to bounding the total degrees. We obtain an algorithm for verifying bounds on the directional degreewidth of the tournament, whose runtime is polynomial time when the total degree is unbounded, or fixed-parameter tractable time with respect to the total degree bound otherwise. We also provide a polynomial-time algorithm for computing a $P_3$-free backedge graph of a tournament, if it exists. Together with existing results, the latter result settles the complexity of determining whether a tournament admits an $H$-free backedge graph when $H$ is any graph on three vertices.