Abstracts

Invited speakers

Federico Ardila - San Francisco State University

TITLE: Combinatorics of CAT(0) cube complexes

ABSTRACT:

A "cube complex" X is a space built by gluing cubes together. We say that X is "CAT(0)" if it has non-positive curvature - roughly speaking, this means that X is shaped like a saddle. CAT(0) cube complexes play an important role in pure mathematics (group theory) and in applications (phylogenetics, robot motion planning). We show that, surprisingly, CAT(0) cube complexes can be described completely combinatorially. This description gives a proof of the conjecture that any d-dimensional CAT(0) cube complex X "fits" in d- dimensional space. It also leads to an algorithm for finding the shortest path between two points in X (and hence to find the fastest way to move a robot from one position to another one). The talk will describe joint work with Megan Owen and Seth Sullivant.

Chris Godsil- University of Waterloo

TITLE: Graph Spectra and Quantum State Transfer

ABSTRACT:

If A is the adjacency matrix of X we define a transition matrix

H(t) := exp( itA );

this is a unitary matrix which defines a so-called continuous quantum walk. I will discuss how questions of physical interest concerning $H$ can be attacked using tools developed in work on eigenvalues of graphs. This has led to new results on quantum state transfer, and to a number of interesting problems in graph theory.

Gregg Musiker  - University of Minnesota

TITLE: Linear Systems on Tropical Curves

ABSTRACT:

A tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system of a divisor on a tropical curve analogously to the classical counterpart. We investigate the structure of such linear systems as a cell complex and show that they are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, these linear systems define maps from tropical curves to tropical projective space. This is joint work with Christian Haase and Josephine Yu.

Rosa Orellana - Dartmouth College

TITLE: Kronecker Coefficients

ABSTRACT:

One of the main open problems in combinatorial representation theory of the symmetric group is to obtain a combinatorial interpretation for the Kronecker coefficients. The Kronecker coefficients are obtained when when we take the tensor product of two irreducible representations of the symmetric group.

This talk is a survey what is known about the Kronecker coefficients and describe some recent results. We will present a closely related family of coefficients called the reduced Kronecker coefficients and their importance in understanding the Kronecker coefficients. In particular, we will discuss the complexity and stability of the Kronecker coefficients. This is joint work with Emmanuel Briand and Mercedes Rosas.

Michael Orrison - Harvey Mudd College

TITLE: Algebraic Voting Theory

ABSTRACT:

In this talk, I'll give an introduction to what might be called "algebraic voting theory." In particular, I'll show how the representation theory of the symmetric group can play a natural and sometimes surprising role when it comes to formulating and answering questions in voting theory. I'll also describe an algebraically motivated extension of the Condorcet criterion that leads to some unexpected results concerning forbidden words of generalized Condorcet winners.

Bernhard Schmidt - Nanyang Technological University

TITLE: Circulant Weighing Matrices of Small and Large Weight

ABSTRACT:

We show that all circulant weighing matrices of weight bounded by a constant arise from what we call "irrducible orthogonal families" which can be enumerated by a finite algorithm. We also describe a new result on circulant weighing matrices of the largest possible weight, i.e., circulant Hadamard matrices, which settles the only known open case of the Barker sequence conjecture.

Catherine Yan - Texas A&M University

TITLE: The Symmetry between Crossings and Nestings in Combinatorial Structures

ABSTRACT:

This talk is a survey of recent progresses on the enumeration of crossings and nestings in combinatorial structures. We describe a new combinatorial model-- the fillings of moon polyominoes, which provides a unified approach to classical combinatorial analysis on permutations, words, matchings, set partitions, multigraphs, and Young tableaux. In the talk we will concentrate on three pairs of combinatorial statistics over the fillings

(1) the longest northeast (NE) and southeast (SE) chains,
(2) the number of NE and SE chains of length 2, and
(3) four families of mixed statistics (to be defined in the talk).

We present enumerative results and show that there is an elegant symmetry between each pair of statistics. These results are connected to many other areas, for example, free probability theory, random matrix theory, representation theory, and mathematical biology.

 

Abstracts of the contributed talks