Combinatorics, Algebra, & Topology Seminar
Spring 2017
All talks are from 12001300 in the Seminar room, unless otherwise specified.

Apr27

Digraphs and toric idealsWalter MorrisGeorge Mason UniversityTime: 12:00 PM
View Abstract
The columns of the nodearc incidence matrix of a connected acyclic directed graph D generate an affine semigroup under addition. If D has m nodes and n arcs, then the nullspace of the nodearc incidence matrix has dimension n − m + 1. The arcs (i, j) of D are associates to monomials which generate a multiplicative semigroup S_D. The kernel of the homomorphism phi:k[u_1,...,u_n] > k[t_1,...,t_m,t_1^(1),...,t_m^(1)] is called the toric ideal I_D. It is known that I_D is generated by binomials (differences of monomials), and that a minimal generating set contains at least n − m + 1 binomials. We characterize connected acyclic digraphs for which ID is generated by exactly n − m + 1 binomials.

Apr20

From hyperelliptic to superelliptic curvesTony ShaskaOakland Univ.Time: 12:00 PM
View Abstract
Hyperelliptic curves are the class of curves better understood among all curves of genus $g \geq 2$. In this talk, we will explore whether the theory of hyperelliptic curves can be extended to superelliptic curves. We will consider both the algebraic geometry and the number theory aspects. In particular, we will focus on automorphisms, equations of curves over , minimal models, etc. The talk will be accessible to a general audience.

Apr06

Quiver representations and generalizations of the pentagon identity, IITime: 12:00 PM
View Abstract
The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver (i.e. orientation of a simplylaced Dynkin diagram). This generalization was first described implicitly via cluster algebras/categories and socalled “wallcrossing” formulas in Donaldson—Thomas theory. (I don’t understand either of those things, so…) In this talk, we describe the identities explicitly by a dimension counting argument. Namely, we calculate the Betti numbers of the equivariant cohomology algebra of the quiver’s representation space in two different ways corresponding to two natural stratifications — an approach suggested by Kontsevich and Soibelman in relation to the Cohomological Hall Algebra of quivers. Time permitting, we may discuss a more advanced generalization corresponding to pairs of Dynkin quivers.

Mar30

Quiver representations and generalizations of the pentagon identityTime: 12:00 PM
View Abstract
The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver (i.e. orientation of a simplylaced Dynkin diagram). This generalization was first described implicitly via cluster algebras/categories and socalled “wallcrossing” formulas in Donaldson—Thomas theory. (I don’t understand either of those things, so…) In this talk, we describe the identities explicitly by a dimension counting argument. Namely, we calculate the Betti numbers of the equivariant cohomology algebra of the quiver’s representation space in two different ways corresponding to two natural stratifications — an approach suggested by Kontsevich and Soibelman in relation to the Cohomological Hall Algebra of quivers. Time permitting, we may discuss a more advanced generalization corresponding to pairs of Dynkin quivers.

Mar23

Polynomials, graphs and cohomology, Or I need help with this problem, interested?Susama AgarwalaTime: 12:00 PM
View Abstract
In this talk, I present a family of graphs representing a family of polynomials. I define a differential graded algebra on this family and discuss barriers to computing the cohomology. Time permitting, I discuss how this is secretly a graphical approach to understanding mixed Tate Motives.

Mar09

Alexander Polynomials of Rational Links, IIMark KidwellTime: 12:00 PM
View Abstract
We will quickly review knots and links in threespace, then give Alexander's 1928 definition of the Alexander polynomial. We will explain how to compute terms of the polynomial by Kauffman's clock moves. We will define rational links, and show how clock moves can be done in a systematic way that demonstrates special properties of their Alexander polynomials. This is joint work with Kerry Luse.

Mar02

Alexander Polynomials of Rational Links, IMark KidwellTime: 12:00 PM
View Abstract
We will quickly review knots and links in threespace, then give Alexander's 1928 definition of the Alexander polynomial. We will explain how to compute terms of the polynomial by Kauffman's clock moves. We will define rational links, and show how clock moves can be done in a systematic way that demonstrates special properties of their Alexander polynomials. This is joint work with Kerry Luse. (Antiabstract: This talk will have nothing to do with matroids.)

Feb23

Matroid KazhdanLusztig PolynomialsTime: 12:00 PM
View Abstract
Abstract: The classical KazhdanLusztig polynomials associated to a Coxeter group have been studied continuously for nearly 40 years with results relating intersection cohomology groups of Schubert varieties to certain bases of the Hecke algebra and to certain combinatorial objects. The relatively new matroid KazhdanLusztig polynomials have striking similarities to the classical case and have potential for developing new connections. We will review the classical case as well as summarize the main results known for the matroid case. If time permits we will discuss a formula for the matroid KazhdanLusztig polynomials in terms of flag Whitney numbers.

Feb16

Pebbling on Graph ProductsFranklin KenterUSNATime: 12:00 PM
View Abstract
Pebbling on graphs is a oneplayer game where the player may move pebbles from one vertex to another by removing another pebble from the same vertex. The pebbling number π(G) is the least number of pebbles required so that regardless the initial configuration of pebbles, a pebble can reach any vertex. Graham conjectured that the pebbling number for the cartesian product, G \box H, is bounded above by π(G)π(H). We show the conjecture is true up to a constant factor. Namely, π(G \box H) ≤ 2π(G)π(H). We extend this technique to other graph constructions as well and discuss possible approaches to Graham's conjecture. This is joint work with John Asplund (Dalton State) and Glenn Hurlbert (VCU).

Feb09

Deltamatroids: Why? Part 3Time: 12:00 PM

Feb02

Deltamatroids: Why? Part 2Time: 12:00 PM

Jan27

Deltamatroids: Why? Part 1Time: 12:00 PM
View Abstract
We introduce deltamatroids, which are related to cellularly embedded graphs and to DNA recombination. We give some new results and highlight some old ones.