ÐÓ°ÉÂÛ̳

Strongly and Robustly Critical Graphs

Abstract: In extremal combinatorics, it is common to focus on structures that are minimal with respect to a certain property. Critical and list-critical graphs occupy a prominent place in chromatic graph theory.  A k-critical graph is a graph with chromatic number k whose proper subgraphs are (k-1)-colorable.  List coloring is a well-known variation on classical vertex coloring where each vertex of a graph receives a list of available colors, and we seek to find a proper coloring of the graph such that the color used on each vertex of the graph comes from the list of colors corresponding to the vertex.  In 2008, Stiebitz, Tuza, and Voigt introduced strongly critical graphs which are graphs that are k-critical yet properly colorable with respect to every non-constant assignment that assigns list of size (k-1) to each vertex in the graph.  

In this talk we show how to construct large families of strongly critical graphs.  We also strengthen the notion of strong criticality by extending it to the framework of DP-coloring which is a generalization of list coloring that has been studied by many researchers since its introduction in 2015.  We call this strengthening of strong criticality robust criticality.  We discuss some basic properties of robustly critical graphs and give a general method for generating large families of such graphs.  As an application of these notions, we show how strong criticality (resp. robust criticality) can be used to give improved bounds on the list chromatic number (resp. DP chromatic number) of certain Cartesian products of graphs.

This is joint work with Anton Bernshteyn, Hemanshu Kaul, and Gunjan Sharma  

Friday, September 6, 2024

Bach Nguyen, Xavier University of Louisiana

Poisson geometry and representation theory of quantum cluster algebra

Abstract: The relationship between Poisson geometry and cluster algebra was first studied by M. Gekhtman, M. Shapiro, and A. Vainshtein. Following their work, we study the global geometry picture of the affine Poisson varieties associated to a cluster algebra and its quantization, root of unity quantum cluster algebra. In particular, we prove that the spectrum of the upper cluster algebra, endowed with the GSV Poisson structure, has a Zariski open orbit of symplectic leaves and give an explicit description of it. Our result provides a generalization of the Richardson divisor of Schubert cells in flag varieties. Further, we describe the fully Azumaya loci of the root of unity upper quantum cluster algebras, using the theory of Poisson orders. This classifies their irreducible representations of maximal dimension. This is a joint work with Greg Muller, Kurt Trampel and Milen Yakimov.

Thursday, April 18, 2024

Jeffrey Mudrock, ÐÓ°ÉÂÛ̳

Counting Packings of List Colorings of Graphs

Abstract: List packing is a relatively new notion that was first suggested by Alon, Fellows, and Hare in 1996, but the suggestion was not formally embraced until a recent paper of Cambie, Cames van Batenburg, Davies, and Kang in 2021.  Given a list assignment for a graph, list packing asks for the existence of multiple pairwise disjoint list colorings of the graph. Several papers have recently appeared that study the existence of such a packing of list colorings.  In this talk we consider counting these packings. 

We define the list packing function of a graph G as the guaranteed number of proper L-packings of a prescribed size over all list assignments L that assign the same number of colors to each vertex of G.  In the case the prescribed size is one, the list packing function of G is equal to its list color function which is the well-studied analogue of its chromatic polynomial in the context of list coloring.  Inspired by the well-known behavior of list color functions and chromatic polynomials, we consider the list packing function of a graph in comparison to its classical coloring counterpart.  This leads to a generalization of a recent theorem of Dong and Zhang (2023), which improved results going back to Donner (1992), about when the list color function of a graph equals its chromatic polynomial. Further, we show how a polynomial method can be used to generalize bounds on the list packing number of sparse graphs to exponential lower bounds on the corresponding list packing functions.

This is joint work with Hemanshu Kaul.

Thursday, March 21, 2024

Steven Clontz, ÐÓ°ÉÂÛ̳

Connectedness of finite topological spaces

Abstract: While much of the theory of general topology focuses on infinite-sized sets, by focusing on finite-sized spaces we can often develop insights that can aid in the general theory. In this talk, we will explore how to encode finite topologies using structures from discrete mathematics, in order to explore the necessity for finite biconnected topological spaces to have a dispersion point.

Thursday, January 11, 2024

Andrei Pavelescu, ÐÓ°ÉÂÛ̳

Hereditary Classes of Graphs

Abstract: Although known to be finite by the Graph Minor Theorem of Robertson and Seymour, the sets of forbidden minors for several classes of spatial graphs are not yet known. The classes considered are (n)-apex graphs, (n)-cap planar graphs, knotlessly embeddable graphs, and apex-nIL graphs. In the absence of complete lists of forbidden minors, it is hard do decide whether a given graph belongs to some of these types of graphs. I shall provide an update on the status of the ongoing searches for these minor minimal graphs and provide a hefty number of open questions.

Thursday, November 16, 2023

Dan Silver, ÐÓ°ÉÂÛ̳

Combinatorial Integration and Graph Coloring

Abstract: A celebrated theorem of De Rham establishes an isomorphism between De Rham cohomology and singular cohomology of 3-manifolds via integration of forms. In joint work with Susan Williams we used an analogous combinatorial integration to relate Fox and Dehn colorings of link diagrams in surfaces. Here we use integration to obtain a well-known bijection between proper 3-colorings of edges and 4-colorings of regions of bridgeless cubic graphs embedded in surfaces.

Thursday, November 9, 2023

Elena Pavelescu, ÐÓ°ÉÂÛ̳

Connected domination in plane triangulations

Abstract: A set of vertices of a graph G such that each vertex of G is either in the set or is adjacent to a vertex in the set is a dominating set of G. If additionally the set of vertices induces a connected subgraph of G, then the set is a connected dominating set of G. The domination number d(G) is the smallest number of vertices in a dominating set of G, and the connected domination number cd(G) is the smallest number of vertices in a connected dominating set of G.  Matheson and Tarjan proved that for triangulations of the plane with n vertices, the domination number is at most n/3. They also conjectured that this upper bound can be improved to n/4. We study connected domination numbers for plane triangulations.  We present some data for graphs with less than 15 vertices,  and we show that Matheson and Tarjan's conjecture cannot be extended to the connected domination number. We conjecture that  for triangulations of the plane with n vertices, the connected domination number is at most n/3.  We show that the difference cd(G)-d(G) can be arbitrarily large. This is joint work with Felicity Bryant. 

Thursday, November 2, 2023

Jeff Mudrock, ÐÓ°ÉÂÛ̳

An Algebraic Approach to Counting Colorings of S-labeled Graphs

Abstract: The notion of S-labeling is a common generalization of classical vertex coloring, signed k-coloring, DP-coloring, group coloring, and coloring of gained graphs that was introduced in 2019 by Jin, Wong, and Zhu.  In this talk we will show a polynomial method for bounding the number of colorings of S-labeled graphs from below.  Our method uses an enumerative version of the Combinatorial Nullstellensatz discovered by Alon and Furedi in 1993.  We also demonstrate how our method can be used to prove new lower bounds on the DP color function, which is the DP analogue of the chromatic polynomial, of graphs satisfying certain sparsity conditions.  This is joint work with Hemanshu Kaul and Samantha Dahlberg. 

Monday, October 16, 2023

Takayuki Watanabe, Chubu University

Numerical experiments on Random Relaxed Newton’s Methods

Abstract: We give some numerical results on Random Relaxed Newton’s Methods which were proposed by Sumi to compute an approximate root of a given polynomial. He proved that this randomized algorithm almost surely works well if a large noise is inserted. In this talk, we demonstrate by numerical experiments that even small noise can make the randomized algorithm successful, and discuss a mathematical conjecture.

Friday,  April 7, 2023

Applications of Homogeneous Fiber Bundles to Schubert Varieties

Abstract: In this talk we will connect the theory of Schubert varieties with the theory of equivariant embeddings by using homogeneous fiber bundles over flag varieties. We will show that the homogenous fiber bundles obtained from Bott-Samelson-Demazure-Hansen varieties are always toroidal. Furthermore, we will show how to identify wonderful varieties among them. We will discuss the proof of a generalization of a conjecture of Hodges and Yong for deciding when a Schubert variety is spherical with respect to an action of a Levi subgroup. If time permits, we will discuss applications of our results to Schubert varieties in general.

This is a joint work with Pinaki Saha from the Indian Institute of Technology in Bombay.

From Unknotted Curves on Seifert Surfaces to Contractible 4-Manifolds

Abstract: In this talk I will explain some basic constructions of 3-manifolds (lens spaces and integral homology spheres) that bound contractible or acyclic 4-manifolds, and state some open questions/conjectures regarding those. The talk will also include some new results which were obtained during a recent REU program at my home institution.

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 C^x acts on V with weights equal to one, then the quotient C^x\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 C^x-principal bundles on flag varieties, the resulting quotient X= C^x\Z for certain C^x-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.

Categorification of Link Invariants VII

Abstract: See below

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.

Categorification of Link Invariants VI

Abstract: See below

Categorification of Link Invariants V

Abstract: See below

Categorification of Link Invariants IV

Abstract: See below

Categorification of Link Invariants III

Abstract: See below

Categorification of Link Invariants II

Abstract: See below

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

Generic Gelfand-Tsetlin Representations of Nonstandard Quantized Orthogonal Algebras

Abstract: We construct infinite-dimensional analogs of classical representations of the nonstandard quantized enveloping algebra U_q'(so_n) by rationalizing the classical formulas from finite-dimensional representations. These also provide representations for the universal enveloping algebra U(so_n) as q approaches 1. We use these new representations to embed U_q'(so_n) into a skew group algebra of shift operators.

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.

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.

Isometric G₂-Manifolds

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.

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.

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.

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.

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.

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.

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.

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.

Knots Groups

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.

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/G^θ, where θ is an involutory automorphism. This is equivalent to analyze K=G^θ 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.

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.

Commuting Varieties and Cohomological Complexity Theory

Abstract: Let k be an algebraically closed field and C_r(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 C_r(n) for n , r ≥ 4. In this talk, we describe, for all r ≥ 7, the irreducible component(s) of maximal dimension of C_r(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.

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.

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.

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.

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.

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).

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 2Ï€ 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.

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.

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.

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.

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 SL_n(Z) or Sp_n(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).

Lucas Congruences

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.

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.

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 Z^d 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.

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.

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.

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.

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.

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.