# Pure Mathematics Seminar

## Current Talks

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

Tuesday, February 1, 2022 at 3:30 p.m. in MSPB 370 | Arik Wilbert, University of South Alabama |
Abstract: The term "categorification" refers to the process of upgrading set-theoretic notions to category-theoretic notions. By categorifying a mathematical object, one often discovers that the original object can be seen as a shadow of a much richer structure. This additional layer of structure often leads to new insights and results about the object of interest. In this lecture series, we will focus on how to use algebraic and representation-theoretic tools to categorify two well-known invariants of knots and links. In the first two lectures we will discuss how Khovanov homology categorifies the Jones polynomial. The Jones polynomial can be realized as a specialization of the HOMFLYPT polynomial. In the last three lectures we will discuss how to categorify the HOMFLYPT polynomial using Hochschild homology of Soergel bimodules. |

Friday, April 8, 2022 | Scott Kaschner, Butler University |
Abstract: TBA |

## Previous Talks

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

Thursday, October 14, 2021 |
Jordan Disch, Iowa State University |
Abstract: We construct infinite-dimensional analogs of classical representations of
the nonstandard quantized enveloping algebra U |

Friday, February 14, 2020 | Melissa Zhang, University of Georgia |
Abstract: Annular links, or links in the solid torus, can help model many objects in low-dimensional topology: they can be used to study the braids and tangles that model the movement of particles in the plane; they can represent transverse links in the standard contact structure in the three-sphere; and they serve as patterns for satellite constructions in 4D topology. It is therefore important to be able to tell different annular links apart. For this, we typically rely on annular link invariants. One way to obtain a homology-type invariant for annular links is to modify such an invariant for links in the three-sphere, by introducing a filtration capturing the presence of a distinguished unknot (sometimes called a "braid axis"). In joint work with Linh Truong, we define an annular filtration on a complex related to Khovanov homology introduced by Sarkar, Seed, and Szabó. From this, we obtain a 2D family of annular link invariants, which we show share many properties and applications with Grigsby, Licata, and Wehrli's annular Rasmussen invariants. |

Friday, October 25, 2019 | Sam Nelson, Claremont McKenna College |
Abstract: Biquandle brackets are a class of invariants of oriented knots and links which includes the classical quantum invariants (Alexander/Conway, Jones, HOMFLYpt, Kauffman polynomials) and the biquandle 2-cocycle invariants as special cases, but include many new invariants as well. In this talk we will see a gentle introduction to biquandle brackets. |

Friday, April 12, 2019 | Christopher Lin, University of South Alabama |
Abstract: G |

Friday, March 8, 2019 | Jared Holshouser, University of South Alabama |
Abstract: In 2017, Vladimir Tkachuk introduced a new selection principle: Say I give you a sequence of open sets, and demand you pick one point from each. Can you make your choices in such a way that the result collection of points is closed and discrete? Studying this property on a space of continuous functions provides a path to understanding the largeness of the underlying space. Over the summer Steven Clontz and I clarified the exact connection between closed discrete selection (with pointwise convergence) and classical selection principles. This past fall Chris Caruvana and I extended these results to the compact open topology. In this talk we will examine this property and its connection to classical selection principles. |

Friday, February 22, 2019 | Steven Clontz, University of South Alabama |
Abstract: Consider the "selection game" for a pair of sets A,B played as follows: during each round Player 1 chooses an element of A, followed by Player 2 choosing an element from 1's choice. At the end of the game, Player 2 wins if the set of their choices belongs to B. Two games are said to be dual if one player has a winning strategy in one game exactly when the opposite player has a winning strategy in the dual. Then a classic result of Galvin proves that the Rothberger selection game, where both A and B are the collection of open covers of a topological space, is dual to the point-open selection game, where A is the collection of local bases for each point of the space and B is the collection of open non-covers. Analogous results have been shown to hold for many other selection games studied in the literature; the presenter will demonstrate a trivially verified sufficient condition that guarantees that duality. |

Friday, November 16, 2018 | Larry Rolen, Vanderbilt University |
Abstract: In this talk, we will discuss a relatively new modular-type object known as a locally harmonic Maass form. We will discuss recent joint work with Ehlen, Guerzhoy, and Kane with applications to the theory of L-functions. In particular, we find finite formulas for certain twisted central L-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. Applications to the congruent number problem will be given. |

Friday, November 9, 2018 | Drew Lewis, University of South Alabama |
Abstract: The notions of ind-varieties and ind-groups (infinite dimensional analogues of algebraic varieties and algebraic groups) were first introduced by Shafarevich some 50 years ago. This structure has seen renewed interest in the last 15 years in the study of automorphism groups of affine varieties. We will give a gentle introduction to these constructs and show how they have been used recently to study automorphism groups of affine varieties. |

Friday, November 2, 2018 | Nemanja Kosovalic, University of South Alabama |
Abstract: We discuss how the problem of symmetric bifurcation of periodic solutions of some nonlinear wave equations in higher spatial dimensions, is intimately related to the group action of the symmetric group on the solution set of certain Diophantine equations, which are sums of squares. This is joint work with Brian Pigott from Wofford College, SC. |

Friday, October 26, 2018 | Daniel Nakano, University of Georgia |
Abstract: Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of a Steinberg module with a simple module with restricted highest weight admits a good filtration. In this talk, I will survey results in this area and present new results where we verify the aforementioned good filtration statement (i.e., Steinberg tensored with restricted simple module) when (i) p ≥ 2h-4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module is corresponding to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five. The talk represents joint work with Christopher Bendel, Cornelius Pillen and Paul Sobaje. |

Friday, October 19, 2018 | Susan Williams, University of South Alabama |
Abstract: Draw a knot in the plane. The part of the knot that is enclosed by a generically placed circle meets the circle in 2n points for some n, and is called an embedded 2n-tangle. Can you conclude anything about the knot - for example, that it is nontrivial - by looking only at one of its embedded tangles? We seek tangle invariants that persist in some fashion as factors of analogous invariants of the knots in which they embed. The case n = 1 is classical. In his 1999 thesis, David Krebes gave a persistent invariant of 4-tangles. This was extended for general n in a paper he wrote with Dan and me while here at South Alabama. Recently, Dan and I found a pleasing short proof of Krebes’s Theorem for n = 2 or 3 using elementary lemmas about spanning trees of graphs. It serves as a nice introduction to our recent and ongoing work on graph-theoretic techniques in knot theory. The talk represents joint work with Daniel Silver. |

Friday, October 5, 2018 | Daniel Silver, University of South Alabama |
Abstract: The fundamental group of a knot complement is an important invariant, a complete invariant of any prime knot. We review basic facts about knot groups, and we describe the two main methods for presenting them, presentations of Wirtinger and Dehn. The latter part of the talk represents joint work with Lorenzo Traldi and Susan Williams. |

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 |

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 | Christopher 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 | Daniel 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. |