Brandeis Combinatorics Seminar -  Spring 2023

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

February 28: In person

Speaker: Ira Gessel (Brandeis)

Title: Problems in enumerative combinatorics

Abstract:  I will discuss some problems in enumerative combinatorics that I have thought about over the years. These involve positive integers with nice generating functions, integer ratios of factorials, graphical enumeration, exponential generating functions, and Bernoulli numbers.

March 7: In person

Speaker: Charles Wang (Harvard)

Title:  Landau-Ginzburg models for cominuscule homogeneous spaces

Abstract:  Landau-Ginzburg (LG) models have been used to study various aspects of mirror symmetry. Rietsch constructed LG-models for homogeneous spaces using Lie-theoretic data, and Rietsch's models have been studied in specific cases using various methods. In these special cases, further study of Rietsch's models was aided by having a concrete LG model defined in terms of (generalized) Plucker coordinates. To generalize these models, we present a type-independent Plucker coordinate construction of Rietsch's LG models for all cominuscule homogeneous spaces using combinatorial properties of the associated fundamental representations. This is joint work in progress with Peter Spacek at TU Chemnitz.

March 14: On Zoom

Speaker: Coleen Robichaux (UCLA)

Title:  Degrees of Grothendieck polynomials and Castelnuovo-Mumford regularity

Abstract:  We give an explicit formula for the degree of a vexillary Grothendieck polynomial. This generalizes a previous result of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt for degrees of symmetric Grothendieck polynomials. We apply these formulas to compute the Castelnuovo-Mumford regularity of certain Kazhdan-Lusztig varieties. We then relate these results to non-intersecting lattice paths. This is joint work with Jenna Rajchgot and Anna Weigandt.

March 21: On Zoom

Speaker:  Matthieu Josuat-Vergès (CNRS, Universite Paris Cite)

Title: Cluster parking functions

Abstract:  Parking functions on one side, the cluster complex on the other wide, are much studied combinatorial objects.  In this work we show that they interact quite well, by introducing cluster parking functions.  I will first motivate the definition, by reviewing parking functions on one side, and the complex of dissections of a polygon (type A cluster complex) on the other side.  Cluster parking functions have a lot of structure: an action of the symmetric group (just like parking functions), they are a simplicial complex (just like the cluster complex).  I will present results about the associated characters of the symmetric group, and the topology of the complex.

March 28: On Zoom

Speaker: Vasu Tewari (University of Hawaii)

Title: Combinatorics of forest polynomials

Abstract:   I will introduce a new basis for the ring of polynomials that we call forest polynomials. This basis is particularly friendly toward reduction modulo the ideal of quasisymmetric polynomials, and as such allows us to describe the cohomology class of the permutahedral variety in terms of Schubert classes via a parking procedure. I will also use this perspective to motivate a multivariate analogue of mixed Eulerian numbers. Joint work with Philippe Nadeau (CNRS & Univ. Lyon 1).

April 18: On Zoom

Speaker: Valentin Feray (CNRS, Université de Lorraine)

Title: Graphon limit and large independent sets in uniform random cographs

Abstract: Cographs are by definition $P_4$-free graphs, i.e. graphs avoiding the path $P_4$ as induced subgraph. In this talk, we will consider a uniform random cograph with $n$ vertices, for large $n$. We shall describe the (random) "graphon" limit of this object, which can be constructed starting from a Brownian excursion. Motivated by some probabilistic work around Erdős-Hajnal conjecture, we will also consider large independent sets in uniform cographs. For both aspects, cographs behave differently from most other $H$-free random graphs.

No background on cographs, graphon convergence or Brownian excursion will be assumed. Based on joint work with F. Bassino, M. Bouvel, M. Drmota, L. Gerin, M. Maazoun and A. Pierrot. 

April 25: In person

Speaker: Gleb Nenashev (Brandeis)

Title:  Cuntz algebras automorphisms: graphs and stability of permutations

Abstract:   Stable permutations are a class of permutations that correspond to a permutative automorphisms of the Cuntz algebra. I will present the necessary background. 

Our main result is a characterization of the stable permutations in terms of two sequences of graphs. As applications I will prove the conjecture that almost all permutations are not stable. Some open problems will be discussed. This is joint work with Roberto Conti and Francesco Brenti.

Links to previous semesters: 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.