# Algebra and Combinatorics Seminar

## Spring 2018

Date: | January 19, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Anton Dochtermann, Texas State University |

Title: | Coparking functions and h-vectors of matroids |

Abstract: | The h-vector of a simplicial complex X is a well-studied invariant with connections to algebraic aspects of its Stanley-Reisner ring. When X is the independence complex of a matroid Stanley has conjectured that its h-vector is a ‘pure O-sequence’, i.e. the degree sequence of a monomial order ideal where all maximal elements have the same degree. The conjecture has inspired a good deal of research and is proven for some classes of matroids, but is open in general. Merino has established the conjecture for the case that X is a cographical matroid by relating the h-vector to properties of chip-firing and `G-parking functions' on the underlying graph G. We introduce and study the notion of a ‘coparking’ function on a graph (and more general matroids) inspired by a dual version of chip-firing. As an application we establish Stanley’s conjecture for certain classes of binary matroids that admit a well-behaved `circuit covering'. Joint work with Kolja Knauer |

Date: | January 26, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Karina Batistelli, U. N. Cordoba |

Title: | Some Lie subalgebras of the matrix quantum pseudo-differential operators |

Abstract: | In this talk, we will characterize the irreducible quasifinite highest weight modules of some subalgebras of the Lie algebra of matrix quantum pseudodifferential operators N x N. In order to do this, we will first give a complete description of the anti-involutions that preserve the principal gradation of the algebra of NxN matrix quantum pseudodifferential operators and we will describe the Lie subalgebras of its minus fixed points. We will obtain, up to conjugation, two families of anti-involutions that show quite different results when n=N and n |

Date: | January 30, 2018 |

Time: | 4:00pm |

Location: | BLOC 220 |

Speaker: | Davis Penneys, The Ohio State University |

Title: | Exotic fusion categories: EH3 exists! |

Abstract: | Fusion categories generalize the representation categories of (quantum) groups, and we think of them as objects which encode quantum symmetry. All currently known fusion categories fit into 4 families: those coming from groups, those coming from quantum groups, quadratic categories, and those related to the extended Haagerup (EH) subfactor. First, I'll explain what I mean by the preceding sentence. We'll then discuss the extended Haagerup subfactor, along with the newly constructed EH3 fusion category (in joint work with Grossman, Izumi, Morrison, Peters, and Snyder), and the possibility of the existence of EH4. |

Date: | February 2, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Daniel Creamer, Texas A&M University |

Title: | A Computational Approach to Classifying Modular Categories by Rank |

Abstract: | Modular categories are of interest in a variety of disciplines stretching from abstract algebra to theoretical physics. It was recently proved by Bruillard, Ng, Rowell, and Wang, that there are a finite number of modular categories given a fixed rank. I present a computer assisted approach to classifying modular categories by their rank. |

Date: | February 9, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Nathan Williams, UT Dallas |

Title: | Fixed Points of Parking Functions |

Abstract: | We define an action of words in [m]^n on R^m to give a new characterization of rational parking functions. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani's zeta map on rational parking functions when m and n are coprime, and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington's sweep map on rational Dyck paths. This is joint work with Jon McCammond and Hugh Thomas. |

Date: | February 10, 2018 |

Time: | 08:00am |

Location: | BLOCKER |

Speaker: | Combinatexas |

Title: | |

Date: | February 11, 2018 |

Time: | 8:00pm |

Location: | BLOCKER |

Speaker: | Combinatexas |

Title: | |

Date: | February 16, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Alex Kunin, Penn State University |

Title: | Hyperplane neural codes and the polar complex |

Abstract: | This talk concerns combinatorial and algebraic questions arising from neuroscience. Combinatorial codes arise in a neuroscience setting as sets of co-firing neurons in a population; abstractly, they record intersection patterns of sets in a cover of a space. Hyperplane codes are a class of combinatorial codes that arise as the output of a one layer feed-forward neural network, such as Perceptron. Here we establish several natural properties of non-degenerate hyperplane codes, in terms of the {\it polar complex} of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a non-degenerate hyperplane code is shellable. Moreover, we show that all currently known properties of hyperplane codes follow from the shellability of the appropriate polar complex. Lastly, we connect this to previous work by examining some algebraic properties of the Stanley-Reisner ideal associated to the polar complex. This is joint work with Vladimir Itskov and Zvi Rosen. |

Date: | February 23, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Ka Ho Wong, Chinese University of Hong Kong |

Title: | Asymptotic expansion formula for the colored Jones polynomial and Turaev-Viro invariant for the figure eight knot |

Abstract: | The volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the $(2N+1)$-th Turaev-Viro invariant for the knot complement $\SS^3 \backslash K$ can be expressed as a sum of the colored Jones polynomial of $K$ evaluated at $\exp(2\pi i/ (N+1/2))$. That leads to the study of the asymptotic expansion formula (AEF) for the colored Jones polynomial of $K$ evaluated at half-integer root of unity. When $K$ is the figure eight knot, by using saddle point approximation, H.Murakami had already found out the AEF for the $N$-th colored Jones polynomial of $K$ evaluated at $\exp(2\pi i/N)$. In this talk, I will first review the strategy Murakami used to prove the AEF of the colored Jones polynomial. Then, I will further discuss, for $M$ with a fixed limiting ratio of $M$ and $(N+1/2)$, how the AEF for the $M$-th colored Jones polynomial for the figure eight knot evaluated at $(N+1/2)$-th root of unity can be obtained. As an application of the asymptotic behavior of the colored Jones polynomials mentioned above, we obtain the asymptotic expansion formula for the Turaev-Viro invariant of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots. |

Date: | March 2, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Shilin Yu, Texas A&M University |

Title: | Quantization and representation theory |

Abstract: | In this talk, I will talk about a geometric way to construct representations of noncompact semisimple Lie groups, though no prior knowledge is required. Kirillov's coadjoint orbit method suggests that (unitary) irreducible representations can be constructed as geometric quantization of coadjoint orbits of the group. Except for a lot of evidence, the quantization scheme meets strong resistance in the case of noncompact semisimple groups. I will give a new perspective on the problem using deformation quantization of symplectic varieties and their Lagrangian subvarieties. This is joint work in progress with Conan Leung. |

Date: | March 23, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Dimitar Grantcharov, UT Arlington |

Title: | Singular Gelfand-Tsetlin Modules and BGG Differential Operators |

Abstract: | Every irreducible finite-dimensional module of the general linear Lie algebra gl(n) can be described with the aid of the classical Gelfand-Tsetlin formulas. The same formulas can be used to define a gl(n)-module structure on some infinite-dimensional modules - the so-called generic (nonsingular) Gelfand-Tsetlin modules. In this talk we will introduce Gelfand-Tsetlin modules and discuss recent progress on the study of singular Gelfand-Tsetlin gl(n)-modules and relations with BGG differential operators. The talk is based on a joint work with V. Futorny, L. E. Ramirez, and P. Zadunaisky. |

Date: | April 13, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Sarah Witherspoon & Catherine Yan |

Title: | Algebra and Combinatorics Spring 2019 Course Discussion. |

Date: | April 13, 2018 |

Time: | 4:00pm |

Location: | BLOC 220 |

Speaker: | Jurij Volcic , Ben-Gurion University of the Negev |

Title: | Joint Algebra and Combinatorics - Linear Analysis seminar. A Nullstellensatz for noncommutative polynomials: advances in determinantal representations |

Date: | April 20, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Aleksandra Sobieska, Texas A&M University |

Title: | Counterexamples for Cohen-Macaulayness |

Abstract: | Let L in Z^n be a lattice, I its corresponding lattice ideal, and J the toric ideal arising from the saturation of L. We produce infinitely many examples, in every codimension, of pairs I,J where one of these ideals is Cohen-Macaulay but the other is not. |

Date: | April 27, 2018 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Xingting Wang, Temple University |

Title: | Noncommutative algebra from a geometric point of view |

Abstract: | In this talk, I will discuss how to use algebro-geometric and Poisson geometric methods to study the representation theory of 3-dimensional Sklyanin algebras, which are noncommutative analogues of polynomial algebras of three variables. The fundamental tools we are employing in this work include the noncommutative projective algebraic geometry developed by Artin-Schelter-Tate-Van den Bergh in 1990s and the theory of Poisson order axiomatized by Brown and Gordon in 2002, which is based on De Concini-Kac-Priocesi’s earlier work on the applications of Poisson geometry in the representation theory of quantum groups at roots of unity. This talk demonstrates a strong connection between noncommutative algebra and geometry when the underlining algebra satisfies a polynomial identity or roughly speaking is almost commutative. |