Pure Mathematics Seminar
|Friday, November 4, 2022 at 2:30 p.m. in MSPB 213||Scott Larson, University of Georgia||
Positivity in Weighted Flag Varieties
Abstract: Let V be a finite dimensional complex vector space and Z the nonzero vectors. If the complexification of the circle Cx acts on V with weights equal to one, then the quotient Cx\Z is projective space. If the weights are changed to any other positive numbers, then the quotient is weighted projective space, which is a familiar object in algebraic geometry and plays a role in geometric invariant theory. Generalizing Z to Cx-principal bundles on flag varieties, the resulting quotient X= Cx\Z for certain Cx-actions on Z is called a weighted flag variety. In general, X can have singularities, but will always be a projective, normal, and rationally smooth orbifold.
William Graham proved a famous result in 2001 of positivity in equivariant cohomology of flag varieties. Abe-Matsumura prove a similar theorem in 2015 for weighted Grassmannian flag varieties, but their parameters are rather mysterious. In joint work with William Graham and Arik Wilbert, we generalize positivity in equivariant cohomology to all weighted flag varieties, and offer a geometric interpretation of all parameters.
|Tuesday, April 12, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants VII
Abstract: See below
|Friday, April 8, 2022||Scott Kaschner, Butler University||
Geometric Limits in Complex Dynamics
Abstract: For maps of one complex variable, f, given as the sum of a degree n power map and a degree d polynomial, I will outline results regarding the limiting dynamics as n approaches infinity. Specifically, I will give a general description of the limiting set of points with bounded orbit. Lastly, I will discuss the generalization of these phenomena to a several variables setting.
|Tuesday, April 5, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants VI
Abstract: See below
|Tuesday, March 29, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants V
Abstract: See below
|Thursday, March 24, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants IV
Abstract: See below
|Tuesday, February 15, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants III
Abstract: See below
|Tuesday, February 8, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants II
Abstract: See below
|Tuesday, February 1, 2022||Arik Wilbert, University of South Alabama||
Categorification of Link Invariants I
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.
Prerequisites: The prerequisite for this series is some basic knowledge about groups, algebras and modules. Some prior exposure to categories and homological algebra will be helpful for the second and third part. No knowledge about topology will be assumed.
Thursday, October 14, 2021
|Jordan Disch, Iowa State University||
Generic Gelfand-Tsetlin Representations of Nonstandard Quantized Orthogonal Algebras
Abstract: We construct infinite-dimensional analogs of classical representations of the nonstandard quantized enveloping algebra Uq'(son) by rationalizing the classical formulas from finite-dimensional representations. These also provide representations for the universal enveloping algebra U(son) as q approaches 1. We use these new representations to embed Uq'(son) into a skew group algebra of shift operators.
|Friday, February 14, 2020||Melissa Zhang, University of Georgia||
Annular Concordance Invariants from Sarkar-Seed-Szabo's Spectral Sequence
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||
Biquandle Brackets: An Introduction
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: G2-manifolds are a class of 7-dimensional manifolds that arise naturally in Riemannian geometry, and have grown very important in physics. We will review basic concepts in G2-geometry, discuss what is meant by these structures being "isometric" - leading up to the result that identifies the space of all such isometric structures as a smooth compact manifold. We will also discuss how this space is immersed topologically into a very important moduli space. If time permits, we will comment on general trends in the field of G2-geometry.
|Friday, March 8, 2019||Jared Holshouser, University of South Alabama||
Closed Discrete Selections of Continuous Functions
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||
Dual Selection Games
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||
Locally Harmonic Maass Forms and Central L-Values
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||
Automorphism Groups of Affine Varieties from the Perspective of ind-Groups
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||
Symmetric Vibrations of Nonlinear Wave Equations
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||
On Tensoring with the Steinberg Representation
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||
Tangles and Knots: A View with Trees
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||
Borel Orbits in Polarization and Lattice Paths
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/Gθ, where θ is an involutory automorphism. This is equivalent to analyze K=Gθ orbits in the flag variety G/B. In fact, their geometry is of importance in the study of Harish-Chandra modules and their closures can be considered as Schubert varieties. The focus of this talk will be on the enumeration problem of Borel orbits in the symmetric space SL(n,C)/S(GL(p,C)xGL(q,C)). We will show that the Borel orbits are parameterized by the lattice paths in a p+1 by q+1 grid moving by horizontal, vertical and diagonal steps weighted by an appropriate statistic. In addition, we will present various t-analogues of the rank generating function for the inclusion poset of Borel orbit closures.
|Friday, November 10, 2017||William Hardesty, Louisiana State University||
Support Varieties for Algebraic Groups and the Humphreys Conjecture
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||
Commuting Varieties and Cohomological Complexity Theory
Abstract: Let k be an algebraically closed field and Cr(n) the variety of commuting r-tuples of nilpotent n x n matrices over k. These commuting varieties have been extensively studied in the case r = 2, however very little is known for large r. One of the challenging problems is to determine irreducible components of Cr(n) for n , r ≥ 4. In this talk, we describe, for all r ≥ 7, the irreducible component(s) of maximal dimension of Cr(n) when k is of characteristic ≠ 2, 3. We also discuss the connection between commuting varieties and cohomological complexity theory of finite group schemes. This is joint work with Paul Levy and Klemen Šivic.
|Friday, November 18, 2016||Lucius Schoenbaum, Louisiana State University||
Cartesian Closed Categories and the Lambda Calculus
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||
Real Graded Division Algebras
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||
Varieties Associated to G-Modules
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||
Lifting Modules for a Finite Group of Lie Type to its Ambient Algebraic Group
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||
Representations of Three-Point Algebras
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,C).
|Friday, January 22, 2016||Abhijit Champanerkar, College of Staten Island and The Graduate Center, CUNY||
Densities and Semi-Regular Tilings
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||
Solitons in G2-Geometry
Abstract: G2-structures are special sub-bundles of the natural frame bundle of a 7-dimensional manifold M. They are special in the sense that they induce irreducible representations of the space of differential forms on M. A special sub-class of G2-structures is called "torsion-free" G2-structures, which acquire more symmetry by ways of parallel translation. These torsion-free G2-structures were fervently sought-after by both mathematicians and physicists due to the extra symmetries that they enjoy. In this talk, we will discuss special solutions (called solitons) to a geometric flow that was proposed as an analytic tool to obtain torsion-free G2-structures. In particular, we will discuss a preliminary classification of these solitons. Towards the end of the talk, we will make comparisons with Ricci-solitons and comment on related questions to some associated moduli spaces.
|Friday, November 20, 2015||Andrei Pavelescu, University of South Alabama||
Commutative Matrix Rings
Abstract: The nilradical of K0 of a commutative ring R can defined to be the intersection of all the kernels of K0(R) -> K0(F) over all maps from R to F with F a field. This definition can be extended to non-commutative rings, provided one knows the structure of maximal commutative matrix subrings. In this talk we are going to explore this topic and look at some examples of such rings.
|Friday, November 13, 2015||Hung Ngoc Nguyen, University of Akron||
The Largest Character Degree and a Conjecture of Gluck
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)2. We confirm this conjecture in the case where |F(G)| is coprime to 6. We also extend the conjecture to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure. This is a joint work with Cossey, Halasi, and Maroti.
|Friday, October 30, 2015||Alexander Hulpke, Colorado State University||
Calculations with Matrix Groups over the Integers
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 SLn(Z) or Spn(Z) of finite index. For determining such an index the structure of classical groups over residue class rings Z/mZ, and the representation theory of classical groups become the major tools.
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 p-adic properties, some of which remain conjectural. A result of Gessel shows that Apéry's sequence satisfies Lucas-type congruences. We prove corresponding congruences for all sporadic Apéry-like sequences. While, in several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences, there is one sequence in particular, often labeled (η), for which we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist-Zudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers. This talk is based on joint work with Amita Malik.
|Friday, October 2, 2015||Cornelius Pillen, University of South Alabama||
Bounding Cohomology for Finite Groups
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||
Periodic Graphs, Spanning Trees and Mahler Measure
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 Zd symmetry. Examples can be found on tiled bathroom floors (d=2) or in crystal structures (d=3).
For such graphs, a Laplacian matrix with polynomial entries can be defined. Its determinant is called the Laplacian polynomial Δ of the graph. The Mahler measure of Δ is the growth rate of the number of spanning trees of increasingly large finite quotients of the graph. We refer to it as the complexity growth rate. Physicists call it the thermodynamic limit, which sounds better.
|Friday, September 11, 2015||Elena Pavelescu, University of South Alabama||
Oriented Matroids and Straight-Edge Embeddings of Graphs
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 K9, the complete graph on nine vertices, contains a non-split link with three components. This talk is based on joint work with Ramin Naimi.
|Friday, September 4, 2015||Scott Carter, University of South Alabama||
Twisted Forests and Algebraic Homology Theories II
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 Yn. The singularities of knotted n-foams correspond to the pictures and descriptions that I gave last time. I'll also go into a bit more detail on how the homology theory is constructed. Finally, I will introduce these braided forests, and attempt to give a categorical interpretation.
|Friday, August 28, 2015||Scott Carter, University of South Alabama||
Twisted Forests and Algebraic Homology Theories I
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||
An Independent Axiom System for the Real Numbers
Abstract: It is well-known that the set R of real numbers is the unique model of the complete ordered field axioms. It is also known that the axioms are not independent. In particular, one can prove the commutative axiom of addition from the other axioms for a ring with identity (this is a common textbook exercise). In this talk, I will show that the completeness axiom is, algebraically, a very strong assumption. In fact, it makes the majority of the algebraic axioms redundant. I will then present an independent axiom system for the reals. This talk should be accessible to all faculty as well as advanced undergraduates.
|Friday, February 6, 2015||Thomas Brüstle, Université de Sherbrooke and Bishop's University, Canada||
On the Non-Leaving-Face Property for Associahedra
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 E6. The key ingredient in the proofs is a normalization, a sort of projection from the associahedron to a face.
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.