# Algebra and Combinatorics Seminar

## Fall 2017

Date: | September 1, 2017 |

Time: | 3:00pm |

Location: | BLOC 506A |

Speaker: | Eric Rowell, Texas A&M University |

Title: | Organizational meeting |

Date: | September 15, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Michael Anshelevich, Texas A&M University |

Title: | Product formulas on posets and Wick products. |

Abstract: | We will construct "incomplete" version of several familiar posets, and prove a product formula on posets. Then we will apply these results to the study of Wick products corresponding to the Charlier, free Charlier, and Laguerre polynomials. For the fourth and perhaps most interesting example of Wick products, I do not know the appropriate poset structure. However their inversion and product formulas can still be obtained by less conceptual techniques. As a consequence, we obtain the formula for the linearization coefficients of the free Meixner polynomials. |

Date: | September 22, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Patrick Brosnan , University of Maryland |

Title: | Hessenberg varieties and a conjecture of Shareshian---Wachs |

Abstract: | I will explain joint work with Tim Chow proving a conjecture of Shareshian---Wachs which relates a combinatorial object, the so-called chromatic symmetric function of a certain graph, to a certain action of the symmetric on the cohomology of a Hessenberg variety first studied by J. Tymoczko. I should mention that, shortly after Chow and I posted our proof to the ArXiv, a completely independent proof relying on a map of Hopf algebras and the theorem of Aguiar---Bergeron---Sottile was posted by M. Guay-Paquet. The Hessenberg varieties in the title are certain smooth subvarieties of the the complete flag variety studied first by the applied mathematicians de Mari and Shayman. They were later generalized by de Mari, Procesi and Shayman to a setting where the general linear group is replaced with an arbitrary reductive group. In this case, Tymoczko's dot action becomes a representation of the Weyl group, and it is an interesting problem to determine this representation. I will present some results in this direction. In particular, I will explain a restriction formula that generalizes Guay-Paquet's proof that his Hopf algebra map respects comultiplication. |

Date: | September 29, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Sarah Witherspoon , Texas A&M University |

Title: | Algebraic deformation theory and the structure of Hochschild cohomology |

Abstract: | Some questions about deformations of algebras can be answered by using Hochschild cohomology, and in particular by using its Lie/Gerstenhaber brackets. Until very recently there was no independent description of this Lie structure for an arbitrary resolution, a big disadvantage both theoretically and computationally. In this talk, we will first introduce Hochschild cohomology and explain its role in algebraic deformation theory. We will then summarize recent progress by several mathematicians, focusing on examples. |

Date: | October 6, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Tian Yang, Texas A&M University |

Title: | Volume conjectures for quantum invariants |

Abstract: | Supported by numerical evidences, Chen and I conjectured that at the root of unity exp(2π i/r) instead of the usually considered root exp(π i/r), the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Wittens invariants (that grow polynomially by the Asymptotic Expansion Conjecture), which may indicate a different geometric interpretation of those invariants than the SU(2) Chern-Simons gauge theory. In this talk, I will introduce the conjecture and show further supporting evidences, including recent joint works with Detcherry and Detcherry-Kalfagianni. |

Date: | October 13, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Yiby Morales, Universidad de los Andes |

Title: | The five-term exact sequence for Kac cohomology |

Abstract: | The group of equivalence classes of abelian extensions of Hopf algebras associated to a matched pair of finite groups was described by Kac in the 60’s as the first cohomology group of a double complex, whose total cohomology is known as the Kac cohomology. Masuoka generalized this result and used it to compute some groups of abelian Hopf algebra extensions. Since Kac cohomology is defined as the total cohomology of a double complex, there is an associated spectral sequence. I will explain how we compute the five-term exact sequence associated to this double complex, which can be used to compute some other groups of abelian extensions. This is joint work with César Galindo. |

Date: | October 20, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Benjamin Schröter, TU-Berlin |

Title: | Multi-splits in hypersimplicies and split matroids |

Abstract: | Multi-splits are a class of coarsest regular subdivisions of convex polytopes. In this talk I will present a characterization of all multi-splits of two types of polytopes, namely products of simplices and hypersimplices. It turns out that the multi-splits of these polytopes are in correspondence with one another and matroid theory is the key in their analysis, as all cells in a multi-split of a hypersimplex are matroid polytopes. Conversely, the simplest case of multi-splits of hypersimplices give rise to a new class of matroids, which we call split matroids. The structural properties of split matroids can be exploited to obtain new results in tropical geometry. |

Date: | October 27, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Laura Colmenarejo, York University |

Title: | Factorization formulas for singular Macdonald polynomials |

Abstract: | We prove some factorization formulas for singular Macdonald polynomials indexed by particular partitions called quasistaircases. We also show some applications of these factorizations and some conjectures we are working on. This is joint work with Charles F. Dunkl and Jean-Gabriel Luque. |

Date: | November 10, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Henry Tucker, UC San Diego |

Title: | Invariants and realizations of near-group fusion categories |

Abstract: | Classification of fusion categories by possible ring structures realized by the tensor product was first considered by Eilenberg-Mac Lane in the case where all objects are invertible under the tensor product, i.e. the Grothendieck ring is a group ring. More recently Tambara-Yamagami, Izumi, Evans-Gannon, and others have established classification results for near-group fusion categories: those with a Grothendieck ring with basis a monoid consisting of a group adjoined with a single non-invertible element. In this talk we will survey the classification results on these categories with an emphasis on the dichotomy between the cases determined by the integrality of the non-invertible object. Specifically, in the integral case it is known that the categories are group-theoretical, and in this case we will discuss quasi-Hopf algebra realizations given by cleft extensions. |

Date: | November 16, 2017 |

Time: | 2:45pm |

Location: | BLOC 628 |

Speaker: | X. Shawn Cui, Stanford University |

Title: | On Two Invariants of Three Manifolds from Hopf Algebras |

Abstract: | We prove a conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury (TVBW) invariant and the Witten-Reshetikhin-Turaev (WRT) invariant, respectively. By a classical result relating TVBW and WRT, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radford invariant for the Drinfeld double of the Hopf algebra. However, whether the relation holds for non-semisimple Hopf algebras has remained open, partly because the introduction of framings in this case makes the Kuperberg invariant significantly more complicated to handle. We give an affirmative answer to this question. An important ingredient in the proof involves using a special Heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the Heegaard diagram. |

Date: | November 17, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | X. Shawn Cui, Stanford University |

Title: | Topological quantum computation and compilation |

Abstract: | Topological quantum computation is a fault tolerant protocol for quantum computing using non-abelian topological phases of matter. Information is encoded in states of multi-quasiparticle excitations(anyons), and quantum gates are realized by braiding of anyons. The mathematical foundation of anyon systems is described by unitary modular tensor categories. We will show one can encode a qutrit in four anyons in the SU(2)_4 anyon system, and universal qutrit computation is achieved by braiding of anyons and one projective measurement which checks whether the total charge of two anyons is trivial. We will also give an algorithm to approximate an arbitrary quantum gate with the ones from the anyon system. The algorithm produces more efficient circuits than the Solovay-Kitaev algorithm. Time allowed, applications in quantum complexity classes will also be addressed. |

Date: | November 30, 2017 |

Time: | 2:45pm |

Location: | BLOC 628 |

Speaker: | Cesar Galindo, Universidad de los Andes |

Title: | Pointed finite tensor categories over abelian groups |

Abstract: | In this talk we will give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application, we will prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group. This talk is base on arXiv:1707.05230, joint work with Iván Angiono. |

Date: | December 1, 2017 |

Time: | 3:00pm |

Location: | BLOC 117 |

Speaker: | Carlos Arreche, UT Dallas |

Title: | Projectively integrable linear difference equations and their Galois groups |

Abstract: | To a linear difference system S is associated a differential Galois group G that measures the differential-algebraic properties of the solutions. We say S is integrable if its solutions also satisfy a linear differential system of the same order, and we say S is projectively integrable if it becomes integrable “modulo scalars”. When the coefficients of S are in C(x) and the difference operator is either a shift, q-dilation, or Mahler operator, we show that if S is integrable then G is abelian, and if S is projectively integrable then G is solvable. As an application of these results one can show certain generating functions arising in combinatorics satisfy no algebraic differential equations. This is joint work with Michael Singer. |