DISCRETE CATS SEMINAR

Discrete CATS Seminar

Speaker:  Zhexiu Tu, Centre College

Title:  Topological Representations of Matroids and the cd-index

Abstract:

There are several different topological representations of non-orientable matroids. In this talk, inspired by Swartz's work, I will show an explicit fully partitioned homotopy sphere d-arrangement S that is a CW-complex whose intersection lattice is the geometric lattice of the corresponding matroid for matroids of rank < 5. Moreover S has a d-sphere in it that is a regular CW-complex. We will also look at enumerative properties, including how the flag f-vector formula of Billera, Ehrenborg and Readdy for oriented matroids applies to arbitrary matroids.

For further information, see Discrete CATS Seminar

Date:
Location:
745 Patterson Office Tower
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Discrete CATS Seminar

Speaker:  Rafael González D'león, Universidad Sergio Arboleda and York University.

Title:  The Whitney dual of a graded poset

Abstract:

Two posets are Whitney duals to each other if the (absolute value of their) Whitney numbers of the first and second kind are switched between the two posets.   We introduce new types of edge and chain-edge labelings of a graded poset which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual and we show how to explicitly construct a Whitney dual using a technique that involves quotient posets. As an application of our main theorem, we show that geometric lattices, the lattice of noncrossing partitions, the poset of weighted partitions studied by González D'León-Wachs and the R*S-labelable posets studied by Simion-Stanley all have Whitney duals. We also show that a graded poset P with a Whitney labeling admits a local action of the 0-Hecke algebra on the set of maximal chains of P. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of P. This is joint work with Josh Hallam (Wake Forest Universtity).

For further information, see Discrete CATS Seminar

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Location:
745 POT
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Discrete CATS Seminar

Speaker:  Radmila Sazdanovic, NC State University

Title:  Chromatic homology theories

Abstract:

This talk is an entree to categorification through knot theory and graph theory. The focal point is the chromatic polynomial and is categorifications: chromatic graph homology over algebra defined by L. Helme-Guizon and Y. Rong, and the homology of a graph configuration space introduced by M. Eastwood, S. Huggett. Time permitting, we will discuss relations between these homology theories in the form of spectral sequences, as well as a new invariant of simplicial complexes inspired by the Eastwood and Huggett approach.

 

For further information, see Discrete CATS Seminar

Date:
Location:
745 POT
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Discrete CATS Seminar

Brian Davis, University of Kentucky.

Regular triangulations and Gröbner bases

In this talk we will give a friendly introduction to regular triangulations, which is a tool for breaking down an integer polytope into simpler pieces: high dimensional triangles!  In the second part of the talk we will present a context in which triangulations make a normally difficult computation much easier.   The main theorem, presented without proof, is really charming!  We assume no knowledge of commutative algebra beyond the prelim sequence.

For more details, see Discrete CATS website

Date:
Location:
745 Patterson Office Tower
Event Series:

Discrete CATS Seminar

Speaker: Richard Ehrenborg, University of Kentucky

Title. Simion's type B associahedron is a pulling triangulation of the Legendre polytope

Abstract: We show that the Simion type B associahedron is combinatorially equivalent to a pulling triangulation of the type A root polytope known as the Legendre polytope. Furthermore, we show that every pulling triangulation of the boundary of the Legendre polytope yields a flag complex. Our triangulation refines a decomposition of the boundary of the Legendre polytope given by Cho.

This is joint work with Gábor Hetyei and Margaret Readdy.

For further information, see Discrete CATS Seminar

Date:
Location:
745 Patterson Office Tower
Event Series:

Discrete CATS Seminar

Title: The $gamma$-coefficients of the tree Eulerian polynomials.

Abstract: We consider the generating polynomial $T_n(t)$ of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the  the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $\T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.
 
Date:
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Location:
POT 745

Discrete CATS Seminar

Speaker: Pamela Harris, Williams College

Title: A proof of the peak polynomial positivity conjecture

 

Abstract: We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive integers, we define $P(S;n)=\{\pi\in\mathfrak{S}_n:P(\pi)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| P(S;n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we introduce a new recursive formula for $|P(S;n)|$ without alternating sums and we use this recursion to prove that their conjecture is true.

Date:
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Location:
POT 745

Discrete CATS Seminar

Speaker: Luis David Garcia Puente, Sam Houston State University
 
Title: What is a sandpile group?
 
Abstract: The theory of sandpile groups started in 1987, when physicists Bak, Tang, and Wiesenfeld created an idealized version of a sandpile in which sand is stacked on the vertices of a (combinatorial) graph and is subjected to certain avalanching rules. The long-term dynamics of this system is encoded by the set of recurrent sandpiles. This set has the structure of a finite

abelian group. This group has been discovered in different contexts and received many names: the sandpile group (statistical physics), the critical group (algebraic combinatorics), the group of

components (arithmetic geometry), and the jacobian of a graph (algebraic geometry).  Algebraically, the sandpile group is isomorphic to the cokernel of the (reduced) Laplacian matrix of the underlying graph. Among many beautiful properties, the order of the sandpile group equals the number of spanning trees of the underlying graph. In this sense, the sandpile group is a more subtle isomorphism invariant of a graph.  In this talk, I will provide an introduction to the subject and showcase a few of my favorite results. Some of these results were obtained in collaboration with students in many undergraduate research projects over the last few years.
Date:
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Location:
POT 745
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