# Pure Mathematics Seminar

### Current Talks

Date | Speaker | Talk |
---|---|---|

Friday, January 19, 2018 | Ozlem Ugurlu, Tulane University |
Abstract: Let G be a complex semisimple algebraic group and B be a Borel subgroup
of G. There are many situations where it is necessary to study the Borel orbits in
G/G |

### Previous Talks

Date | Speaker | Talk |
---|---|---|

Friday, November 10, 2017 | William Hardesty, Louisiana State University |
Abstract: I will begin by introducing the notion of the support variety of a module over a finite group scheme. This will be followed by a brief overview of classical results and calculations for the case when the finite group scheme is the first Frobenius kernel of a reductive algebraic group G. In 1997, J. Humphreys conjectured that the support varieties of indecomposable tilting modules for G (a very important class of modules) is controlled by a combinatorial bijection, due to G. Lusztig, between nilpotent orbits and a certain collection of subsets of the affine Weyl group called "canonical cells". This later became known as the "Humphreys conjecture". I will discuss some recent developments concerning this conjecture, including its complete verification for G = GL(n) (appearing in my thesis) as well as some additional results in other types appearing in joint work with P. Achar and S. Riche. |

Friday, October 27, 2017 | Nham Ngo, University of North Georgia |
Abstract: Let k be an algebraically closed field and C |

Friday, November 18, 2016 | Lucius Schoenbaum, Louisiana State University |
Abstract: During the 1960's and 1970's, connections between logic and category theory were discovered through the work of Lawvere, Lambek, Benabou, Grothendieck, and others. In the 1980's, these developments began to have an impact on many areas of computer science, such as programming language semantics and the design of functional programming languages. In this talk, I will introduce this area but focus on cartesian closed categories and the (simply-typed) lambda calculus, which are related via the Curry-Howard-Lambek correspondence (I will explain what this is). I will also discuss how recent work can be used to widen the purview of this theory. Prerequisites: No category theory other than a basic idea of what categories and functors are. |

Friday, April 15, 2016 | Yuri Bahturin, Memorial University of Newfoundland, Canada and Vanderbilt University |
Abstract: An important step in the classification of group gradings on simple algebras is the determination of graded division algebras. In this talk I will classify simple graded division algebras over algebraically closed field as well as over the field of real numbers and mention consequences of this results for the classification of gradings on real simple associative algebras of finite dimension, as well as on some infinite-dimensional algebras. |

Friday, March 11, 2016 | Paul Sobaje, University of Georgia |
Abstract: Let G be a semisimple algebraic group over an algebraically closed field of characteristic p, and let N be normal subgroup scheme of G. Given a finite dimensional G-module V, the N-submodules of V are permuted by the action of G on V. In this way, one obtains G-varieties which live inside various Grassmannian varieties of V. We will introduce these varieties, study some of their geometric properties, and then discuss applications to the representation theory of G. |

Friday, February 26, 2016 | Cornelius Pillen, University of South Alabama |
Abstract: Let p be a prime and q a power of p. The algebraic closure of the field with p elements is denoted by k. A Zariski-closed subgroup G of the general linear group with entries in k, is an algebraic group. If we replace the field k by a finite field with q elements we obtain a finite group of Lie type, sitting inside the infinite group G. We are interested in the following question: Can a module of the finite group be lifted to a module for the algebraic group? For example, a well-known result of Robert Steinberg says that all the simple modules can be lifted. But in general the answer to the aforementioned question is no. This talk is a survey of known results together with some explicit SL(2) examples. |

Friday, February 12, 2016 | Elizabeth Jurisich, College of Charleston |
Abstract: I will introduce the definition of the three-point algebra and introduce
two field representations for this algebra. We provide a natural free field realization
in terms of a beta-gamma system and the oscillator algebra of the three-point affine
Lie algebra when g=sl(2, |

Friday, January 22, 2016 | Abhijit Champanerkar, College of Staten Island and The Graduate Center, CUNY |
Abstract: For a hyperbolic knot or link K the volume density is the ratio of hyperbolic volume to crossing number, and the determinant density is the ratio of 2π log(det(K)) to the crossing number. We explore limit points of both densities for families of links approaching semi-regular biperiodic alternating links. We explicitly realize and relate the limits for both using techniques from geometry, topology, graph theory, dimer models, and Mahler measure of two-variable polynomials. This is joint work with Ilya Kofman and Jessica Purcell. |

Friday, December 4, 2015 | Chris Lin, University of South Alabama |
Abstract: G |

Friday, November 20, 2015 | Andrei Pavelescu, University of South Alabama |
Abstract: The nilradical of K |

Friday, November 13, 2015 | Hung Ngoc Nguyen, University of Akron |
Abstract: Let F(G) and b(G) respectively denote the Fitting subgroup and the largest
degree of an irreducible complex character of a finite group G. A well-known conjecture
of D. Gluck claims that if G is solvable then |G : F(G)| ≤ b(G) |

Friday, October 30, 2015 | Alexander Hulpke, Colorado State University |
Abstract: For matrix groups over the integers, reduction by a modulus m is a fundamental
algorithmic tool. I will investigate how it can be used to study such groups on the
computer, to test finiteness or finite index. Particular emphasis is given to Arithmetic
groups, that is subgroups of SL This is joint work with A. Detinko and D. Flannery (both NUI Galway). |

Friday, October 16, 2015 | Armin Straub, University of South Alabama |
Abstract: Apéry-like numbers are special integer sequences, going back to Beukers
and Zagier, which are modelled after and share many of the properties of the numbers
that underlie Apéry's proof of the irrationality of ζ(3). Among their remarkable properties
are connections with modular forms and a number of |

Friday, October 2, 2015 | Cornelius Pillen, University of South Alabama |
Abstract: An old conjecture due to Guralnick says the following: There exists a universal bound C such that for any finite group G and any faithful, absolutely irreducible G-module V the dimension of the first cohomology group is bounded above by C. In this talk we give a survey of recent results related to Guralnick's conjecture. We are particularly interested in finding bounds for cohomology groups of finite groups of Lie type. |

Friday, September 25, 2015 | Dan Silver, University of South Alabama |
Abstract: The task of counting spanning trees of a finite graph is happily solved
using the Laplacian matrix and determinants. Not content to leave a good thing alone,
we consider infinite graphs with cofinite free For such graphs, a Laplacian matrix with polynomial entries can be defined. Its determinant
is called the |

Friday, September 11, 2015 | Elena Pavelescu, University of South Alabama |
Abstract: Matroid theory is an abstract theory of dependence introduced by Whitney
in 1935. It is a natural generalization of linear (in)dependence. Oriented matroids
can be thought of as combinatorial abstractions of point configurations over the reals.
To every linear (straight-edge) embedding of a graph one can associate an oriented
matroid, and the oriented matroid captures enough information to determine which pairs
of disjoint cycles in the embedded graph are linked. In this talk, we will introduce
the basics of oriented matroids. Then we show that any linear embedding of K |

Friday, September 4, 2015 | Scott Carter, University of South Alabama |
Abstract: I will try and give a little bit more motivation for these ideas by discussing
the possibility of knotted n-dimensional foams. An n-foam is a space that is locally
modeled upon neighborhoods of points in the space Y |

Friday, August 28, 2015 | Scott Carter, University of South Alabama |
Abstract: Motivated by some of the Reidemeister moves for knotted trivalent graphs, I will describe an algebraic structure that consists of two binary operations. One is associative; the other is self-distributive; the self-distributive operation also distributes over the associative operation, and an additional property holds. Under these conditions, a homology theory is defined that recognizes singularities of knotted foams. There are relations to higher categorical structures, and partially ordered sets. |

Friday, April 24, 2015 | Greg Oman, University of Colorado at Colorado Springs |
Abstract: It is well-known that the set |

Friday, February 6, 2015 | Thomas Brüstle, Université de Sherbrooke and Bishop's University, Canada |
Abstract: D. Sleator, R. Tarjan and W. Thurston showed in 1988 that the associahedron
satisfies the non-leaving-face property, that is, every geodesic connecting two vertices
stays in the minimal face containing both. Recently, C. Ceballos and V. Pilaud established
the non-leaving-face property for generalized associahedra of types B, C, D, and some
exceptional types including E We use methods from cluster categories to define such a normalization, which allows us to establish the non-leaving-face property at once for all finite cases that are modelled using cluster categories, namely the Dynkin diagrams. This talk reports on joint work with Jean-François Marceau. |