| Date | Speaker | Title |
|---|---|---|
| September 15 | Jianping Pan (North Carolina State University) | A bijection between K-Kohnert diagrams and reverse set-valued tableaux |
| Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (RSVT) rule for Lascoux polynomials and reverse semistandard Young tableaux (RSSYT) rule for key polynomials. Besides, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with RSSYT. Ross and Yong introduced K-Kohnert diagrams, which are analogues of Kohnert diagrams. Ross and Yong conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between RSVT and K-Kohnert diagrams. | ||
| September 22 | Logan Crew (Waterloo) | A graph-theoretic approach to plethysms of symmetric functions |
|
The plethysm operation f[g] of two symmetric functions is of foundational interest in algebraic combinatorics. On the representation theoretic side, it corresponds to the composition of symmetric group class functions under the Frobenius characteristic. On the symmetric function side, plethysms occur frequently as one of the most natural ways to describe operators and mappings in the space of symmetric functions.
In this talk, I will give an overview of the fundamentals of plethysm, and give a new combinatorial interpretation for any plethysm by interpreting it in terms of a signed sum over proper colorings of acyclically oriented graphs. I will demonstrate that this graph-theoretic interpretation unifies previous results and gives rise to new plethystic identities. This is based on joint work with Sophie Spirkl. | ||
| September 29 | social hour! | |
| October 6 | Jean-Philippe Labbé (Université du Québec) | Lineup polytopes and applications in quantum physics |
|
To put it simply, Pauli's exclusion principle is the reason why we can't walk
through walls without getting hurt. Pauli won the Nobel Prize in Physics in 1945
for the formulation of this principle. A few years later, this principle received
a geometrical formulation that is still overlooked today. This formulation uses the
eigenvalues of certain matrices (which represent a system of elementary particles,
for example electrons). These eigenvalues form a symmetric geometric object
obtained by cutting a hypercube: it is a hypersimplex.
To represent systems of particles with a non-zero temperature, it is necessary to generalize the hypersimplex to obtain what is called "lineup polytopes". These polytopes are defined using classical notions of combinatorics and discrete geometry. Moreover, they produce new exclusion principles which refine Pauli's principle that shall be put to the test by experimentalists. During this talk, we will see the history behind the introduction of these polytopes and give a presentation of some properties. This is joint work with physicists Julia Liebert, Christian Schilling and mathematicians Eva Philippe, Federico Castillo and Arnau Padrol. | ||
| October 13 | Reading week | no seminar |
| October 20 | Sheila Sundaram | Quasisymmetric functions, descent sets, immaculate tableaux, and 0-Hecke modules |
|
The first half of this talk will be expository and devoted to a discussion of
(quasi)symmetric functions and tableaux.
We define new families of quasisymmetric functions, in particular the new basis of row-strict dual immaculate functions, with an associated cyclic, indecomposable 0-Hecke algebra module. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution ψ on the ring Qsym of quasisymmetric functions. We uncover the remarkable properties of the immaculate Hecke poset induced by the 0-Hecke action on standard immaculate tableaux, revealing other submodules and quotient modules, often cyclic and indecomposable. As in the dual immaculate case, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, defined by a specific descent set. We complete the combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets. We show that the generating functions of all the possible variations of tableaux are characteristics of these 0-Hecke modules, captured in the immaculate Hecke poset. This talk is based on joint work with Elizabeth Niese, Stephanie van Willigenburg, Julianne Vega and Shiyun Wang. | ||
| October 27 | Anna Pun (CUNY-Baruch College) | A raising operator formula for Macdonald polynomials |
|
In this talk, I will give a brief introduction on Catalanimal, a tool that helps us to prove the shuffle theorem under any line, the extended delta conjecture and the Loehr- Warrington conjecture. I will then focus on its variant "Macanimal" which gives us an explicit raising operator formula for the modified Macdonald polynomials. Our method just
as easily yields a formula for an infinite series of GLl characters which truncates to the modified Macdonald polynomials.
This is a joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse and George Seelinger. | ||
| November 3 | Jeremy Chizewer (Waterloo) | The Hat Guessing Number of Graphs (in person) |
|
The hat guessing number HG(G) of a graph G on n vertices is defined in terms of the following game: n players are placed on the n vertices of G, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
In this talk, I will begin with an illustrative example and then show the lower bound on HG(G(n,1/2)), where G(n,1/2) denotes the random graph on n vertices where each edge is included uniformly and independently with probability 1/2. I will also discuss the linear hat guessing number. This is based on joint work with Noga Alon. | ||
| November 10 | Theo Douvropoulos (University of Massachusetts at Amherst) | Recursions and Proofs in Coxeter-Catalan combinatorics |
|
The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.
The W-non-nesting parking functions are relatively well understood; they form the so called algebraic W-parking-space which has a concrete interpretation as a quotient ring. The W-non-crossing ones on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory, representation theory, and combinatorics). One of the main open problems in the field since the early 2000's has been to give a type-independent proof of the W-isomorphism between the algebraic and the non-crossing W-parking spaces. In this talk, I will present such a proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular the enumeration of certain flats of full support via Crapo's beta invariant, the W-Laplacian matrices for reflection arrangements and, in collaboration with Matthieu Josuat-Verges, the refined f- to h- transformation between the cluster complex and the non-crossing lattice of W. | ||
| November 17 | Josip Smolcic (Waterloo) | Algorithms for analytic combinatorics in several variables (in person) |
| In this presentation we will see how to apply the theory of complex analysis to study multivariate generating series by looking at several examples. Specifically, given a rational bivariate generating function G(x, y)/H(x, y) with coefficients f_{i, j} the objective is algorithmically determine asymptotic formulas to approximate f_{rn, sn} as n goes to infinity, for fixed positive integers r and s. In this presentation we demonstrate two approaches for determining the asymptotic formulae, each of which involve determining so-called minimal critical points of the denominator H(x, y) in the direction (r, s). The first approach uses numerical methods to solve systems of polynomial equations which depend on the given bivariate generating function to determine minimal points of the denominator, while the second involves analyzing a map h from the zero-set of H to the real numbers, known as a height map. Software developed for both of these purposes will be demonstrated. | ||
| November 24 | Michael Borinsky (ETH Zurich) | Asymptotics of the Euler characteristic of Kontsevich's
commutative graph complex
|
| I will present results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. By a work of Chan-Galatius-Payne, these results imply the same asymptotic growth rate for the top-weight Euler characteristic of M_g, the moduli space of curves, and establish the existence of large amounts of unexplained cohomology in this space. This asymptotic growth rate follows from new generating functions for the edge-alternating sum of graphs without odd automorphisms. I will give an overview on this interaction between topology and combinatorics and illustrate the combinatorial and analytical tools that were needed to obtain these generating functions. | ||
| December 1 | Sergey Yurkevich (University Paris-Saclay) | Algebraicity of solutions of functional equations with one catalytic variable |
| Numerous combinatorial enumeration problems reduce to the study of functional equations which can be solved by a uniform method introduced by Bousquet-Mélou and Jehanne in 2006. In my talk, I will first briefly explain this result and its proof. Then I will present a new generalization of it to the case of systems of functional equations with one catalytic variable. The method is constructive and yields an algorithm for computing the minimal polynomials of interest.
The talk is based on joint work with Hadrien Notarantonio. | ||
| December 8 | Student talks: Tia Ruza and Kimia Shaban | (in person) |
|
Tia's title: Multivariate Limit Theorems via Analytic Combinatorics in Several Variables
Abstract: Analytic combinatorics in several variables is a field of study focused on the derivation of limit behaviours of multivariate sequences. In this talk, I will provide a variety of examples of applying an automated version of a local central limit theorem. Included in these examples will be a family of permutations with restricted cycles, integer compositions with tracked summands and n-colour compositions with tracked summands. The proof of this automated local central limit theorem will also be briefly discussed, using techniques from the theory of analytic combinatorics in several variables. Kimia's title: An introduction to Coxeter groups and the properties of their weak order Abstract: Coxeter groups, such as the symmetric group of permutations are groups generated by reflections. In this talk, I will discuss the poset structure defined as the weak order of Coxeter groups and introduce the Sperner property. While it is known the weak order of the symmetric group of permutations is strongly Sperner, this talk will focus on extending this result for all finite Coxeter groups. Solving the 1-Sperner case using a bipartite matching algorithm to find the maximum size antichain will be explained. I will define the construction of root systems by Coxeter groups of type A_n, B_n and D_n and discuss its applications towards the order of the Weyl groups listed. | ||