Discrete Math Seminar

Discrete CATS Seminar

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.

Date:
Location:
POT 745

Discrete CATS Seminar

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.

Date:
Location:
POT 745

Discrete CATS Seminar

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.

Date:
-
Location:
POT 745
Tags/Keywords:

Discrete CATS Seminar

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.

Date:
-
Location:
POT 745
Tags/Keywords:

Discrete CATS Seminar

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.

Date:
-
Location:
POT 745

Discrete CATS Seminar

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.

Date:
-
Location:
POT 745

Discrete CATS Seminar

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.
Date:
-
Location:
POT 745

Discrete CATS Seminar

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.
Date:
-
Location:
POT 745