Events

Title: The Matchings Polynomial

Abstract:

A $k$-matching in a graph is a matching of size $k$, and $p(X,k)$ denotes the number of $k$-matchings in $X$.

The matching polynomial of a graph is a form of generating function for the sequence $(p(X,k))_{k\ge0}$.

If is closely related to the characteristic polynomial of a graph. I will discuss some of the many interesting properties of this polynomial, and some of the related open problems.

Title: The multispecies TAZRP and modified Macdonald polynomials

Abstract:

Recently, a formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies asymmetric simple exclusion process (ASEP) on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model.

Title: Finding twin smooth integers for isogeny-based cryptography

Abstract:

Efficient and secure instantiations of cryptographic protocols require careful parameter selection. For the isogeny-based cryptographic protocol B-SIDH, a variant of the Supersingular-Isogeny Diffie Hellman (SIDH) key exchange, one needs to find two consecutive B-smooth integers of cryptographic size such that their sum is prime. The smaller the smoothness bound B is, the more efficient the protocol becomes. This talk discusses a sieving algorithm to find such twin smooth integers that uses solutions to the Prouhet-Tarry-Escott problem.

Title: Promotion and rowmotion – an ocean of notions

Abstract: 

Dynamical Algebraic Combinatorics studies objects important in algebraic combinatorics through the lens of dynamical actions. In this talk, we give a flavor of this field by investigating ever more general domains in which the actions of promotion on tableaux (or tableaux-like objects) and rowmotion on order ideals (or generalizations of order ideals) correspond. This is based on joint works with J. Bernstein, K. Dilks, O. Pechenik, C. Vorland, and N. Williams.

Title: Probabilistic Aspects of Voting, Intransitivity, and Manipulation

Abstract:

Marquis de Condorcet, a French philosopher, mathematician, and political scientist, studied mathematical aspects of voting in the eighteenth century. Condorcet was interested in studying voting rules as procedures for aggregating noisy signals and in the paradoxical nature of ranking  3 or more alternatives. We will begin with a quick survey of some of the main mathematical models, tools, and results in this theory and discuss some recent progress in the area. 

Title: Cospectral Vertices and Isospectral Reductions

Abstract:

Understanding cospectral vertices in graphs is fundamental to understanding what the spectrum of the adjacency matrix can tell us about a graph.  Furthermore, cospectral vertices are necessary in constructions of graphs exhibiting perfect quantum state transfer.  I will talk about how to recognize cospectral vertices via a tool from network dynamics: the isospectral reduction of a graph.  I will explore possible ways of getting new constructions of cospectral vertices by looking at isospectral reductions.

Title: Solving Prellberg and Mortimer's conjecture - bijection(s)
between Motzkin paths and triangular walks

Abstract:

In these difficult times, what we need to feel better is some colorful and elegant bijections.

This talk introduces the work we did with Andrew Elvey-Price (Tours, France) and Irène Marcovici (Nancy, France). Together we answered an open question from Mortimer and Prellberg, asking for a bijection between a family of walks inside a bounded triangular domain (think about a large equilateral triangle subdivided in several smaller equilateral triangles) and the famous Motzkin paths, but which have bounded height.

Title: Mixed Integer Programming - Strength of adding integer variables

Abstract:

Mixed Integer Programming is the problem of optimizing a multi-variate function over some domain constraints where some variables are required to take integer values. From a complexity-theoretic perspective,  problems with fewer integer variables are easier to solve. However, this is not always the case in practice.  We will discuss how performance can be improved when adding integer variables in the context of cutting planes and branch and bound. We will compare several frameworks for doing so in both the context of converting lifting integer and continuous variables to more variables.  We will conclude with recent work on mixed-integer quadratic programming and mention some computational results.

*Note different start time

Title: Real Chromatic Roots of Graphs

Abstract:

In February 1988, I arrived at C&O Waterloo for a postdoc with the late Ron Read. He handed me a paper by Beraha, Kahane and Weiss, and told me to apply it to determining the location of the complex roots of chromatic polynomials.  I’ve returned to the topic every few years since then, with varying degrees of success---some positive results, but still many open problems and conjectures remain.

Title: Chain decompositions for q,t-Catalan numbers

Abstract:

The q,t-Catalan numbers Cat_n(q,t) are polynomials in q and t that reduce to the ordinary Catalan numbers when q=t=1. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Work of Garsia, Haglund, and Haiman has given us combinatorial formulas for Cat_n(q,t) as sums of Dyck vectors weighted by area and dinv. This talk narrates our ongoing quest for a bijective proof of the notorious symmetry property Cat_n(q,t)=Cat_n(t,q).