# CMU Algebra and Combinatorics Seminar

## Fall 2015

### Meeting Times

Wednesdays, 4–5pm, in Pearce 223.

### Schedule

 Date Speaker Affiliation Title 9/30 Lisa DeMeyer Central Michigan University Graph constructions associated to a semigroup 10/14 Gabriel Feinberg Haverford College Geometry, representation theory, and voting 10/21 Travis Scrimshaw University of Minnesota Mathematical physics through combinatorics 10/28 Samantha Dahlberg Michigan State University Hopf Algebras and the antipode of $\text{NCSym}$ 11/11 (3:30-4:30pm) Andrei Minchenko Weizmann Institute The Galois group of a parameterized linear differential equation 11/11 (4:30-5:30pm) Kenneth Barrese University of California, San Diego $p$,$q$ analogues of $m$-level rook numbers 12/2 Emily Gunawan University of Minnesota Cluster Algebras from Surface Triangulations 12/9 Mark Pengitore Purdue University TBD

### Abstracts

Speaker: Lisa DeMeyer (Central Michigan University)
Title: Graph constructions associated to a semigroup
Abstract: Given a commutative semigroup $S$ with $0$, the zero divisor graph $\Gamma(S)$ is a graph whose vertices are labeled by the non-zero zero divisors of $S$ and two distinct vertices are adjacent precisely when the product of the corresponding elements in $S$ is $0$. This construction has been widely studied over the past 20 years. Elementary aspects of the relationship between $S$ and $\Gamma(S)$ will be discussed. Recent and not-so-recent results on the clique homology of $\Gamma(S)$ will be presented. The talk will conclude with a brief mention of other fruitful graph constructions associated to $S$ in the current literature.

Speaker: Gabriel Feinberg (Haverford College)
Title: Geometry, representation theory, and voting
Abstract: How much impact does the voting method have on the outcome of an election? Given some common-sense desired characteristics for a system, can we find an "ideal" election process? The answers to these questions, while potentially troubling, are established. In this talk, I'll introduce a geometric approach of Donald Saari, which leads naturally to these answers. We'll then see how a voting procedure can be considered as a $\mathbb{Q}S_n$-module homomorphism. Following the work of Michael Orrison et al., we'll exploit the well-studied representation theory of the symmetric group to make further observations.

Speaker: Travis Scrimshaw (University of Minnesota)
Title: Mathematical physics through combinatorics
Abstract: We look at various aspects of statistical mechanics that can be explained through combinatorics. In particular, we will look at Kashiwara's crystal basis theory, objects called rigged configurations, and, if time permits, solitons (also known as ultra-discrete box-ball systems).

Speaker: Samantha Dahlberg (Michigan State University)
Title: Hopf Algebras and the antipode of $\text{NCSym}$
Abstract: Consider the algebra $\mathbb{Q} \ll x \gg$ of formal power series in non-commuting variables ${\bf x} = \{x_1, x_2,\dots \}$. The algebra of symmetric functions in non-commuting variables, $\text{NCSym}$, contains functions $f \in \mathbb{Q} \ll x \gg$ such that for all permutations $\pi$, $f(x_1,x_2,\dots) = f(x_{\pi(1)},x_{\pi(2)},\dots)$. In this talk we will introduce Hopf algebras, antipodes and $\text{NCSym}$ as a Hopf algebra. A cancellation-free antipode formula was found by Baker-Jarvis, Bergeron, and Thiem. We also have a formula which appears in a different form and uses Takeuchi's formula and a sign-reversing involution. This is a technique which has been introduced by Benedetti and Sagan.

Speaker: Andrei Minchenko (Weizmann Institute)
Title: The Galois group of a parameterized linear differential equation
Abstract: We will start with the definition and basic properties of the parameterized differential Galois group. It is an important problem of differential Galois theory to build an algorithm that computes this group starting from a differential equation with parameters. An algorithm for the non-parameteric case was found by Hrushovski in 2002, while the parametric case remains unsolved. I will recall recent results with A. Ovchinnikov and M. Singer that give partial solution to this question, and will finish by explaining an approach to solve the problem in general.

Speaker: Kenneth Barrese (University of California, San Diego)
Title: $p$,$q$ analogues of $m$-level rook numbers
Abstract: This talk presents joint work with Nicholas Loehr, Jeffrey Remmel, and Bruce Sagan. The $m$-level rook placements are a generalization of ordinary rook placements. By factoring the m-level rook polynomials of Ferrers boards, it is possible to partition them into equivalence classes. There is a $p,q$-analogue of the $m$-level rook numbers. We have a bijective proof that two boards with the same $m$-level rook numbers have the same $q$-analogues of their $m$-level rook numbers. Surprisingly, they may not have the same $p$-analogues.

Speaker: Emily Gunawan (University of Minnesota)
Title: Cluster Algebras from Surface Triangulations
Abstract: Cluster algebras, introduced by Fomin and Zelevinsky in 2000, link together diverse fields of study, e.g. discrete dynamical systems, Riemann surfaces, representation theory of quivers, knot theory, etc. Cluster algebras are commutative algebras which are generated by a distinguished set of (usually infinitely many) generators, called cluster variables. Starting from a finite set $x_1, x_2,\dots, x_n$, the cluster variables can be computed by an iterated elementary process. They miraculously turn out to always be Laurent polynomials in $x_1, x_2, \dots, x_n$, with positive coefficients. Finding a closed-form formula for the cluster variables is one of the main problems in the theory of cluster algebra. In this talk, I will discuss such a closed-form formula for the class of cluster algebras which can be modeled after triangulations of orientable Riemann surfaces with marked points (my running examples will be a pentagon and an annulus). The formula is given in terms of paths (called $T$-paths) along the edges of a fixed triangulation. The $T$-paths can be used to give a combinatorial proof for a natural basis (consisting of elements which are indecomposable and positive in some sense) for some types of cluster algebras.

Speaker: Mark Pengitore (Purdue University)
Title: TBD
Abstract: Coming soon …