Combinatorics, Algebra, & Topology Seminar

Spring 2017

All talks are from 1200-1300 in the Seminar room, unless otherwise specified.

• Apr

  27
• Digraphs and toric ideals

  Walter Morris

  George Mason University

  Time: 12:00 PM

  View Abstract

  The columns of the node-arc 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 node-arc 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.

• Apr

  20
• From hyperelliptic to superelliptic curves

  Tony Shaska

  Oakland 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.

• Apr

  06
• Quiver representations and generalizations of the pentagon identity, II

  Justin Allman

  Time: 12:00 PM

  View Abstract

  The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver (i.e. orientation of a simply-laced Dynkin diagram). This generalization was first described implicitly via cluster algebras/categories and so-called “wall-crossing” 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.

• Mar

  30
• Quiver representations and generalizations of the pentagon identity

  Justin Allman

  Time: 12:00 PM

  View Abstract

  The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver (i.e. orientation of a simply-laced Dynkin diagram). This generalization was first described implicitly via cluster algebras/categories and so-called “wall-crossing” 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.

• Mar

  23
• Polynomials, graphs and cohomology, Or I need help with this problem, interested?

  Susama Agarwala

  Time: 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.

• Mar

  09
• Alexander Polynomials of Rational Links, II

  Mark Kidwell

  Time: 12:00 PM

  View Abstract

  We will quickly review knots and links in three-space, 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.

• Mar

  02
• Alexander Polynomials of Rational Links, I

  Mark Kidwell

  Time: 12:00 PM

  View Abstract

  We will quickly review knots and links in three-space, 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. (Anti-abstract: This talk will have nothing to do with matroids.)

• Feb

  23
• Matroid Kazhdan-Lusztig Polynomials

  Max Wakefield

  Time: 12:00 PM

  View Abstract

  Abstract: The classical Kazhdan-Lusztig 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 Kazhdan-Lusztig 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 Kazhdan-Lusztig polynomials in terms of flag Whitney numbers.

• Feb

  16
• Pebbling on Graph Products

  Franklin Kenter

  USNA

  Time: 12:00 PM

  View Abstract

  Pebbling on graphs is a one-player 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).

• Feb

  09
• Delta-matroids: Why? Part 3

  Carolyn Chun

  Time: 12:00 PM

• Feb

  02
• Delta-matroids: Why? Part 2

  Carolyn Chun

  Time: 12:00 PM

• Jan

  27
• Delta-matroids: Why? Part 1

  Carolyn Chun

  Time: 12:00 PM

  View Abstract

  We introduce delta-matroids, which are related to cellularly embedded graphs and to DNA recombination. We give some new results and highlight some old ones.