Future students

Abstract: Positroids are a subclass of matroids born in the study of the non-negative Grassmanian by Postnikov in 2006. Since then, there have been a plethora of combinatorial objects indexing positroids, two of these being the families of decorated and bicolored permutations, which are generalizations of classical permutations. These two families can be used to study properties of positroids, and as a byproduct we end up with useful ways to describe a group action on a deck of cards. In this context, we give a definition of invariants under this group action allowing us, as an application, to develop new magic tricks with unusual ways of shuffling cards.

AbstractPositroids were introduced by Postnikov in 2006 as a special class of matroids with nice combinatorial properties. Since 2008, starting with Suho Ho, several attempts have been made to describe the poset of quotients for this class of matroids in a combinatorial way. However, these descriptions are incomplete and always come from the same perspective. That is why we will explore new combinatorial objects and the context in which they arise (magic, polytopes, and antisymmetric algebras) to see if it is possible to describe this poset.

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

Abstract: A tournament is an orientation of a complete graph, and in terms of structural questions, tournaments are often a natural analogue of graphs. Neumann-Lara, in 1982, defined what it means to colour a tournament. Only recently, in 2023, Aboulker, Aubian, Charbit and Lopes defined what the clique number of a tournament is — but it is a bit more complicated than in graphs. What can we say about these parameters, from a structural and computational point of view? There are a few things to say, including a recent joint result with Logan Crew, Xinyue Fan, Hidde Koerts, and Ben Moore.

[Many of you know the severe consequences that a COVID infection has had for my family. I would consider it a kindness if attendees would wear masks. I will provide free masks for those who may need them.]

Abstract:Motivated by the representation theory of finite-dimensional algebras, we recently investigated the notions of left modularity and extremality in (completely) semidistributive lattices. For lattices of torsion classes, we obtain a simultaneous characterization of left modularity and extremality in terms of the behavior of certain indecomposable modules, called bricks. Our results extend the classical theory beyond the realm of finite lattices, while remaining within the framework of (completely) semidistributive lattices. Time permitting, I will also discuss extensions of these results to arbitrary infinite lattices that are completely semidistributive and weakly atomic. This talk is based on recent joint work with Sota Asai, Osamu Iyama, and Charles Paquette.

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

Abstract: A common strategy in many proofs and algorithms is to begin by decomposing a graph into more highly connected pieces. Decomposition is easy when the goal is to obtain connected or 2-connected pieces, and decomposition into 3- or 4-connected pieces is also straightforward in many settings. For higher levels of connectivity, however, no effective and widely applicable notion of decomposition is currently known. To address this, Robertson and Seymour introduced tangles, which capture the k-connected regions of a graph without decomposing. 

Abstract:In a widely circulated manuscript from the 1990s, I.G. Macdonald introduced certain higher-rank analogs of the classical hypergeometric functions $_pF_q$, which are expressed as explicit series in Jack and Macdonald polynomials in one and two sets of variables. For special choices of parameters, these series reduce to the hypergeometric functions of matrix argument introduced earlier by C. Herz and A.T. James, which have numerous applications in number theory, multivariate statistics, signal processing, and random matrix theory.

The classical hypergeometric functions are solutions to the hypergeometric differential equation. Macdonald raised the problem of providing an analogous characterization for higher-rank functions by means of differential equations. Over the years, this problem was solved for a small number of cases where p and q are at most 3. However, as the operators become increasingly complicated, the general problem remained open for 40 years. In this talk, we will present a complete solution. This is joint work with Hong Chen.

There will be a pre-seminar at 1:30pm in MC 6029 in a flipped classroom format based on Macdonald’s manuscript on hypergeometric functions (https://arxiv.org/abs/1309.4568). Participants are expected to read the manuscript in advance, and the session will focus on questions and discussion led by the speaker.

Abstract: The set of dimers (aka perfect matchings) of a connected bipartite plane graph G is a distributive lattice, as shown by Propp. The order relation on this lattice comes from the "height" of a dimer, which is a vector of nonnegative integers. In this talk, I'll focus on the dimer face polynomial of G, which is the height generating function of all dimers of G. This polynomial has close connections to knot invariants on the one hand, and cluster algebras on the other. I'll discuss joint work with Mészáros, Musiker and Vidinas in which we explore these connections. No knowledge of knot theory or cluster algebras will be assumed.

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