Discrete Math Seminar

Discrete CATS Seminar

Title: Mutation of friezes

 

Abstract: Frieze is an array of positive integers satisfying certain rules.  Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras.   Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras.   In this talk, we will discuss friezes of types A and D and their mutations.

 

Date:
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Location:
POT 745

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:
-
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:
-
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:
-
Location:
POT 745

Discrete CATS Seminar

Title: Kazhdan-Lusztig polynomials of thagomizer graphs Abstract: To a graph G, one can associate a polynomial with non-negative integer coefficients called the Kazhdan-Lusztig polynomial of G. More generally, you can obtain the Kazhdan-Lusztig polynomial of any matroid, but today we will focus on the specialization to graphs. The Kazhdan-Lusztig theory for matroids was developed in analogy with the classical theory for Coxeter groups, though there are some important differences which I will touch on lightly. In this talk, we will construct the defining recursion for the Kazhdan-Lusztig polynomial of thagomizer graphs and use this obtain a closed form for the coefficients of the polynomial. No prior knowledge of matroids or Kazhdan-Lusztig polynomials will be assumed.

Date:
-
Location:
POT 745

Discrete CATS Seminar

Title: “Cold atoms, SU(N) symmetry and Young tableaux”

Abstract:  Symmetries play a foundational role in our understanding of physics.
 It is often the case that unexpected symmetries can emerge from unexpected places.
I will discuss a particular example in systems of cold atoms that realize SU(N) symmetry,
where N can be as large as ~10.  I will then describe how these systems can be treated
numerically by using results on the symmetric group dating back to Young.

Date:
-
Location:
POT 745

Discrete CATS Seminar

Discrete CATS Seminar

2pm, Monday Feb 13

POT 745

Speaker: Laura Escobar, UIUC

Title: Resolutions of singularities of Schubert and Richardson varieties
 
Abstract: In the first part of this talk I will present a combinatorial model describing  resolutions of singularities of Schubert varieties, namely tilings by 2-dimensional zonotopes. This is based on joint work with Pechenik, Tenner and Yong. We will then discuss resolutions of singularities for Richardson varieties and talk about the moment polytope of these varieties. In particular, I will give a description of the toric variety of the associahedron.
Date:
-
Location:
POT 745

Discrete CATS Seminar

Discrete CATS Seminar

Monday, Feb 6
2pm
CB 345
 
Speaker: Cyrus Hettle
 
Title: Universal partial words
 
Abstract: A universal word for an alphabet A and a positive integer n is a word containing each of the words of length n over A as a substring exactly once. For instance, de Bruijn sequences are (cyclic) universal words. Universal partial words, introduced by Chen, Kitaev, Mutze, and Sun in 2016, allow for a wild-card character \diamond, which can stand for any letter in the alphabet. We settle and strengthen conjectures posed in the same paper where this notion was introduced. For non-binary alphabets, we show that universal partial words have periodic structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for a family of universal partial words over alphabets of even size. This talk is based on joint work with Bennet Goeckner, Corbin Groothius, Brian Kell, Pamela Kirkpatrick, Rachel Kirsch, and Ryan Solava.
 

 

Date:
-
Location:
CB 345

Discrete CATS Seminar

Speaker: Rafael Gonzalez D'Leon
Title: The boolean algebra and the exterior algebra
Abstract: The boolean algebra is one of the most fundamental objects in the theory of partially ordered sets. Similarly the exterior algebra is one of the most fundamental objects in the theory of associative algebras. I will present a beautiful relation between these two objects that is a good illustration of the interplay between algebra, topology and combinatorics. The talk will also serve as an introduction to the general area of poset topology.

 

Date:
-
Location:
POT 745
Event Series:
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