DOUBLE HEADER, PART II, 1:45 - 2:30 pm

Speaker:  Yannic Vargas, TU Graz

Title:  Hopf algebras, species and free probability

Abstract:

Free probability theory, introduced by Voiculescu, is a non-commutative probability theory where the classical notion of independence is replaced by a non-commutative analogue ("freeness"). Originally introduced in an operator-algebraic context to solve problems related to von Neumann algebras, several aspects of free probability are combinatorial in nature. For instance, it has been shown by Speicher that the relations between moments and cumulants related to non-commutative independences involve the study of non-crossing partitions. More recently, the work of Ebrahimi-Fard and Patras has provided a way to use the group of characters on a Hopf algebra of "words on words", and its corresponding Lie algebra of infinitesimal characters, to study cumulants corresponding to different types of independences (free, boolean and monotone). In this talk we will give a survey of this last construction, and present an alternative description using the notion of series of a species.

Yannic Vargas is visiting Martha Yip.

Tea in POT 745 at 3:14 pm = π time

Speaker:  Mihai Ciucu, Indiana University

Title:  Cruciform regions and a conjecture of Di Francesco

Abstract:

The problem of finding formulas for the number of tilings of lattice regions goes back to the early 1900's, when MacMahon proved (in an equivalent form) that the number of lozenge tilings of a hexagon is given by an elegant product formula. In 1992, Elkies, Kuperberg, Larsen and Propp proved that the Aztec diamond (a certain natural region on the square lattice) of order n has 2^n(n+1)/2 domino tilings. A large related body of work developed motivated by a multitude of factors, including symmetries, refinements and connections with other combinatorial objects and statistical physics. It was a model from statistical physics that motivated the conjecture which inspired the regions we will discuss in this talk.

In recent work on the twenty vertex model, Di Francesco was led to a conjecture which states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted T_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions T_n, we construct a family of cruciform regions C_m,n^a,b,c,d generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of the region T_n is a divisor of the number of tilings of the cruciform region C_2n-1,2n-1^n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Speaker:  Steven Karp, Notre Dame

Title:  q-Whittaker functions, finite fields, and Jordan forms

Abstract:

The q-Whittaker symmetric function associated to an integer partition is a q-analogue of the Schur symmetric function. Its coefficients in the monomial basis enumerate partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the q-Burge correspondence. In the q -> 0 limit, we recover a known description of the classical Burge correspondence (also called column RSK). We use the q-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.

This is joint work with Hugh Thomas.

Part I of DOUBLE HEADER!

745 POT; 1-1:45 pm

Speaker:  Daniel Tamayo, Université Paris-Saclay

Title:  On some recent combinatorial properties of permutree congruences
of the weak order

Abstract:

Since the work of Nathan Reading in 2004, the field of lattice
quotients of the weak order has received plenty of attention on the
combinatorial, algebraic, and geometric fronts. More recently, Viviane
Pons and Vincent Pilaud defined permutrees which are combinatorial
objects with nice combinatorial properties that describe a special
family of lattice congruences. In this talk we will give a brief
introduction into the world of (permutree) lattice congruences, how they
lead to structures such as the Tamari and boolean lattice, followed by
connections to pattern avoidance, automata and some examples of sorting
algorithms.

Daniel Tamayo is visiting Martha Yip.

Qualifying Exam

Speaker:  Williem Rizer, University of Kentucky

Title:  Combinatorics of the Positroidal Stratification of the Totally Nonnegative Grassmannian

Abstract:

The Grassmannian has been an object of much interest in algebra, geometry, and combinatorics. We can decompose the Grassmannian into matroid strata, in which each element of the stratum has the same set of nonzero Plücker coordinates corresponding to the bases of a matroid. If we restrict to the elements of the Grassmannian with nonngeative coordinates, the corresponding matroids are called positroids. Postnikov revealed a family of combinatorial objects that can be used to parameterize positroidal cells. Recently, various authors have studied objects (X-diagrams and LACD colored permutations) that are in correspondence with this family, though the bijections exhibited do not commute with one another. In this talk we discuss all of these objects, the bijections between them, the information they reveal about positroids, and the possible connections and generalizations we can make to fold these newer objects neatly into the family.

Presenting the KOI Combinatorics Lectures.

http://www.ms.uky.edu/~readdy/KOI

Friday, March 31, 2023
03:15 - 03:50 pm Coffee/Tea
04:00 - 05:00 pm Mihai Ciucu, Colloquium, Cruciform regions and a conjecture of Di Francesco, CB 214

Saturday, April 1, 2023 (CB 114)
09:00 - 10:00 am, Arrival/Registration/Meet and Greet
09:59 - 10:00 am Welcome, Welcome speech
10:00 - 11:00 am, Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
11:00 - 11:30 am, Coffee Break
11:30 - 12:30 pm, Eric Katz, Models of matroids
12:30 - 02:30 pm, Lunch Break
02:30 - 03:00 pm, Problem Session, run by Saúl A. Blanco
03:00 - 03:30 pm, Tea time and the One Picture/One Theorem Poster Session
04:00 - 05:00 pm, Richard Ehrenborg, Sharing pizza in n dimensions
06:00 - 08:00 pm, Conference Dinner

Speaker:  Galen Dorpalen-Barry, Ruhr-Universität Bochum

Title:   The Poincaré-extended ab-index

Abstract:

Motivated by a conjecture of Maglione-Voll from group theory, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial.  For posets admitting R-labelings, we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione-Voll's conjecture as well as a conjecture of the Kühne-Maglione. We also recover, generalize, and unify results from Billera-Ehrenborg-Readdy, Ehrenborg, and Saliola-Thomas.

This is joint work with Joshua Maglione and Christian Stump.

We will be meeting in 745 POT and the speaker will be live from Germany via Zoom.

Our website:  https://www.ms.uky.edu/~readdy/Seminar

PhD Defense

Speaker:  William Gustafson

Advisor:  Margaret A. Readdy

Title:  Lattice Minors and Eulerian Posets

Speaker:  Ana Garcia Elsener, Universidad Nacional de Mar del Plata

Title:  skew-Brauer graph algebras

Abstract:

Brauer graph algebras are defined by combinatorial data based on graphs:

Underlying every Brauer graph algebra is a finite graph, the Brauer graph, equipped with with a cyclic orientation of the edges at every vertex and a multiplicity function. This combinatorial data encodes much of the representation theory of Brauer graph algebras and is part of the reason for the ongoing interest in this class of algebras. A known result by Schroll states that Brauer graph algebras, with multiplicity function one, give rise to all possible trivial extensions for gentle algebras. On the other hand, Geiss and de la Peña studied a generalization of gentle algebras called skew-gentle algebras.

In our ongoing project we establish the right definition of skew-Brauer graph algebra in such a way that the result by Schroll can be enunciated in this context. That is, A is a skew-Brauer graph algebra with multiplicity function equal to one if and only if it is the trivial extension of a skew-gentle algebra. Moreover, the family of skew-Brauer graph algebras with arbitrary multiplicity function generalizes the family of Brauer graph algebras with arbitrary multiplicity function.

(Joint work with Victoria Guazzelli from Universidad Nacional de Mar del Plata, Argentina, and Yadira Valdivieso Diaz from Universidad de Puebla, México)

Ana Garcia Elsener is visiting Khrystyna Serhiyenko.

Speaker:  Gábor Hetyei, UNC Charlotte

Title:  Brylawski's tensor product formula for Tutte polynomials of colored graphs

Abstract: 

The tensor product of a graph and of a pointed graph is obtained by replacing each edge of the first graph with a copy of the second. In his expository talk we will explore a colored generalization of Brylawski's formula  for the Tutte polynomial of the tensor product of a graph with a pointed graph and its applications.  Using Tutte's original (activity-based) definition of the Tutte polynomial we will provide a simple proof of Brylawski's formula. This can be easily generalized to the colored Tutte polynomials introduced by Bollobás and Riordan. Consequences include formulas for Jones polynomials of (virtual) knots and for invariants of composite networks in which some major links are identical subnetworks in themselves.

All results presented are joint work with Yuanan Diao, some of them are also joint work with Kenneth Hinson. The relevant definitions and the fundamental results used will be carefully explained.  

G. Hetyei will be a visitor of R. Ehrenborg and M. Readdy the first week of March.