Title: Quasi-cluster algebras and triangulations of the Möbius strip



Abstract: In this talk, we will first define triangulations of marked surfaces, then use it to define quasi-cluster algebras (the equivalent of cluster algebras for non-orientable surfaces). We will list a few properties of these algebras, then we will count the number of triangulations of the Möbius strip, the only surface with a finite number of triangulations.

Title: Rational links represented by reduced alternating diagrams


Abstract: In knot theory, a rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a nonalternating form and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued  fraction expansion of the rational number in which all partial denominators are even (called all-even form}). In this talk we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or equivalently the nonalternating form of the corresponding rational number.    



This is joint work with Yuanan Diao and Claus Ernst. The talk will define and explain all terms, with a general audience in mind.

Title: The Coxeter Transformation and Rowmotion for cominuscule posets

Abstract: There is a combinatorial action, called the Rowmotion, defined on cominuscule posets. It is well-known that this action has order 'h' on the order ideal poset of a cominuscule poset where h is the Coxeter number of the corresponding root system. Also, we will talk about the action of the Coxeter transformation on the order ideals of cominuscule posets. We prove that the Coxeter transformation is periodic of order 'h+1' (up to a sign) in most cases. We will demonstrate combinatorial similarities of the orbits of these two actions.

Title: The Universal Valuation of Coxeter Matroids 



Abstract: Valuations on a family of polytopes are functions which behave nicely with respect to subdivisions in this family. One important question is the determine the structure of the set of all valuations on a certain family. This can be done by constructing a "universal valuation" which is a valuation that can be specialized to any other valuation on this family. Coxeter matroids are a generalization of matroids to an arbitrary root system. As with usual matroids, we can interpret Coxeter matroids as polytopes. In this talk, we will construct a universal valuation for the family of Coxeter matroid polytopes.

Title: Chip firings and the scramble number 

Abstract: Chip firing can be described as a game we play by placing poker chips on the vertices of a graph. The scramble number is a new graph invariant that helps inform us as to how many chips we need to "win" that game. We will explore the properties of the scramble number as well as techniques used to compute it. This is joint work with Dave Jensen.

Speaker:  Ben Reese, University of Kentucky

Title:  Zero-One Schubert Polynomials

Schubert polynomials were first defined by Bernstein-Demazure in 1973-1974, and they arise in several areas of combinatorics and representation theory. We present three different descriptions of these polynomials due to Lascoux-Schutzenberger, Billey-Jockusch-Stanley, and Magyar. We then discuss the Fink-Meszaros-St. Dizier pattern avoidance characterization of Schubert polynomials having zero-one coefficients.  This is motivated by those Schubert polyomials that equal the integer point transform of the generalized permutahedron.

This is a Master exam talk.

Title: Lee-Lee Conjecture on geometric description of c-vectors. 

Abstract: A quiver is a directed graph without oriented 2 cycles and loops.  It has been shown that for an acyclic quiver, the set of non-initial d-vectors (related to the cluster algebra associated to the quiver), the set of positive c-vectors (from a framed quiver), and the set of real Schur roots (from the root system associated to the quiver) coincide. None of these objects are geometric though. In an effort to give a geometric description, Kyu-Hwan Lee and Kyungyong Lee conjectured that the set of roots obtained from a non-self-crossing admissible curve coincide with the set of c-vectors for an acyclic quiver. In this talk, I will describe these objects and talk about this conjecture for the acyclic quivers of finite case.

Title: BOGO Sale

Abstract: This talk will consist of two 20 minute talks which I will be giving at the Joint Math Meetings. Feedback will be solicited.



2:00 - 2:20 Title: k-matching sequences of simplicial complexes.



Abstract: The homotopy type of the matching complex, M_1(G), has been studied for paths, cycles, and trees. In this talk we will generalize 1-matching complex to k-matching complexes, denoted M_k(G) and consider the sequence (M_1(G), M_2(G),…,M_n(G)) up to homotopy for perfect caterpillar graphs.



2:25 - 2:45 Title: A positivity phenomenon in Elser’s Gaussian-cluster percolation model



Abstract: Veit Elser proposed a random graph percolation model in which physical dimension appears as a parameter. From this model, numerical graph invariants els_k(G) , called Elser numbers, naturally arise and Viet Elser conjectured that els_k(G) \geq 0 for all graph $G$ and integers k \geq 2. In this talk we will interpret the Elser numbers as Euler characteristics of nucleus (simplicial) complexes and prove Elser’s conjecture. This is joint work with Galen Dorpalen-Barry, Cyrus Hettle, David Livingston, Jeremy Martin, George Nasr, and Hays Whitlatch.

Title: An introduction to quiver mutations and green sequences.

Abstract: Quiver mutation is an operation one can define on a directed graph that has shown to model the behavior of a large variety of mathematical objects including algebra, topology, physics and number theory. More specifically it arises when studying Teichmuller theory, total positivity of matrices, and even the behavior of high energy particle impacts. We will introduce the process of quiver mutation, and the proceed to explore quivers for a special sequence of mutations called maximal green sequences. The aim of the talk is to discuss recent work that allows one to build maximal green sequences for larger quivers by looking at "component preserving" sequences on induced subquivers. These new techniques have allowed us to construct maximal green sequences for large families of quivers where their existence was previously unknown.

Speaker:  William Gustafson

Title:  Shellability of the uncrossing partial order on matchings

Abstract:

We introduce the uncrossing partial order on matchings, which first arose as the face posets of stratified spaces of planar resistor networks. We discuss its relationship to the type A Bruhat order and give an EC-shelling due to Patricia Hersch and Richard Kenyon.