Brandeis Combinatorics Seminar -  Fall 2024

The Brandeis Combinatorics Seminar is an introductory seminar for combinatorics.
The talks should be (at least partially) understandable to first year graduate students.

            The zoom link is  https://brandeis.zoom.us/j/94622483750

October 29:  The Combinatorics Seminar is replaced by a Division of Arts and Science talk in Abelson 131 (2:20-3:40pm)

Speaker: Pamela Harris (U Wisconsin)

Title:  The Role of Undergraduate Research Experiences in Developing a Sense of Belonging

November 5: 

Speaker: Yuhan Jiang (Harvard)

Title:  The Ehrhart h*-polynomials of positroid polytopes

Abstract: 
A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. 

Positroid polytopes are matroid polytopes of positroids.

In particular, they are lattice polytopes.

The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope.

The Ehrhart series is the generating function of the Ehrhart polynomial, which is a rational function with the numerator called the h*-polynomial.

We compute the h*-polynomials of an arbitrary positroid polytope by a family of shelling orders of it.

We also compute the h*-polynomial of any positroid polytope with some facets removed and we relate it to the descents of permutations.

Our result generalizes that of Early, Kim, and Li for hypersimplices.

November 12: 

Speaker: Sarah Brauner (Brown University)

Title:  Spectrum of random-to-random shuffling in the Hecke algebra

Abstract:  The eigenvalues of a Markov chain determine its mixing time. In this talk, I will describe a Markov chain on the symmetric group called random-to-random shuffling whose eigenvalues have surprisingly elegant—though mysterious—formulas. In particular, these eigenvalues were shown to be non-negative integers by Dieker and Saliola in 2017, resolving an almost 20 year conjecture.

In recent work with Ilani Axelrod-Freed, Judy Chiang, Patricia Commins and Veronica Lang, we generalize random-to-random shuffling to a Markov chain on the Type A Iwahori Hecke algebra, and prove combinatorial expressions for its eigenvalues as a polynomial in q with non-negative integer coefficients. Our methods simplify the existing proof for q=1 by drawing novel connections between random-to-random shuffling and the Jucys-Murphy elements of the Hecke algebra.

November 19: 

Speaker: Sasha Pevzner (Northeastern University) 

Title:   Symmetric group fixed quotients of polynomial rings and Stanley--Reisner rings

Abstract:  Consider the standard action of the symmetric group on a polynomial ring in n variables. We consider the quotient module, called the fixed quotient, obtained by setting each polynomial equal to its images under the action. This is a module over the ring of symmetric polynomials, with relatively little known about its structure. We investigate the collection of these modules as n varies, focusing on a stability pattern in their minimal free resolutions. We state results and conjectures on this stability, then switch gears to study an analogous quotient of a certain Stanley--Reisner ring. Inspired by work of Garsia and Stanton in the 80s, as well as the theory of Hodge algebras put forth by DeConcini, Eisenbud, and Procesi, we develop a framework for bounding the behavior of the polynomial ring fixed quotient in terms of that of the Stanley--Reisner ring.

November 26: 

Speaker: Houcine Ben Dali (Harvard)

Title: Differential equations for the series of hypermaps with control on their full degree profile  

Abstract: We consider generating series of hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that under some particular specializations, these series satisfy the celebrated KP equations in the orientable case, and BKP equations in the non-orientable one. In this talk, I present a family of differential equations which characterizes the full generating series of hypermaps.

I will give a first proof which works for a one parameter deformation of the series of hypermaps related to Jack polynomials. This proof is based on a differential formula for Jack characters obtained in a joint work with Maciej Dołęga. I will also present a combinatorial proof for the orientable case.

December 3: 

Speaker: Isaac Berger (Brandeis)

Title:  Hurwitz numbers for the two-equal-cycles partitions λ=(n,n) and transportation polytopes

Abstract:  We calculate the generating function for arbitrary genus Hurwitz numbers on the cycle types  λ= (n,n), extending work of Chapuy-Stump using the Frobenius lemma.  Apart from the explicit calculation, we also show how to relate the resulting function with the h-vector of a central transportation polytope. The proof involves techniques from representation theory, along with many aspects of combinatorics.

Links to previous semesters: Spring2024, Fall2023, Spring2023, Fall 2022, Spring 2022, Fall 2021, Spring 2020, Fall 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013.