DISCRETE CATS SEMINAR

Discrete CATS Seminar

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.

 

Date:
Location:
745 POT

KOI Combinatorics Lectures

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

Date:
Location:
CB 114

KOI Combinatorics Lectures

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

Date:
Location:
CB 114

Discrete CATS Seminar

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

Date:
Location:
745 POT
Event Series:

Discrete CATS & Algebra Seminar

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.

Date:
Location:
745 POT

Discrete CATS & Algebra Seminar

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.

Date:
Location:
745 POT

Discrete CATS Seminar

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.

Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Speaker:  Marta Pavelka, U Miami

Title:  2-LC triangulated manifolds are exponentially many

Abstract:

We introduce "t-LC triangulated manifolds" as those triangulations obtainable from a tree of d-simplices by recursively identifying two boundary (d-1)-faces whose intersection has dimension at least d - t - 1. The t-LC notion interpolates between the class of LC manifolds introduced by Durhuus-Jonsson (corresponding to the case t = 1), and the class of all manifolds (case t = d). Benedetti-Ziegler proved that there are at most 2^(N d^2) triangulated 1-LC d-manifolds with N facets. Here we show that there are at most 2^(N/2 d^3) triangulated 2-LC d-manifolds with N facets.

We also introduce "t-constructible complexes", interpolating between constructible complexes (the case t = 1) and all complexes (case t = d). We show that all t-constructible pseudomanifolds are t-LC, and that all t-constructible complexes have (homotopical) depth larger than d - t. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen-Macaulay.

This is joint work with Bruno Benedetti. Details of the proofs and more can be found in our paper of the same title.

Marta Pavelka is a student of Bruno Benedetti.

 

Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Speaker:  Thomas McConville, Kennesaw State University

Title:  Lattices on shuffle words

Abstract:

The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle. 

 

Thomas McConville will be visiting Khrystyna Serhiyenko

Date:
Location:
745 POT
Event Series: