FURSP 2018

The Program

The Fields Undergraduate Summer Research Program (FUSRP) welcomes carefully selected undergraduate students from around the world for a rich mathematical research experience in July and August, and I was fortunate enough to be on selected. By the end program we gave presentation presenting our results and submitted a report that would later become a publication.

Besides math the program provides students with several opportunities to socialize with other, network with prospective graduate programs and firms, and explore the city of Toronto.

The Math

I was on a team of five individuals, authors listed in the paper below, advised by Dr. Angèle Foley, from Wilfrid Laurier University. A brief summary of our project is below but broadly speaking we studied algebraic combinatorial objects called chromatic symmetric functions which are associated to graphs.

Project Summary ( from webpage, written by Dr. Foley)

"Chromatic symmetric functions—the focus of this project in combinatorics—sit at the intersection of graph theory and enumeration. These symmetric functions, defined in 1995 by Richard Stanley of MIT, generalize chromatic polynomials, well-known objects in graph theory that count the number of colorings of a particular graph. By contrast, the chromatic symmetric function of a graph is like a super chromatic polynomial—it not only counts the colorings, it counts the number of vertices of each color. This facilitates deeper knowledge of the structure of the graph and allows the exploitation of the machinery of classical symmetric function theory.

Symmetric functions are a long-standing part of algebraic combinatorics, and a fundamental question in symmetric function theory is whether a particular symmetric function, such as the chromatic symmetric function of a given graph, can be expressed with positive coefficients in terms of either the elementary or Schur symmetric function basis. The so-called e-positivity or Schur positivity of a graph is an interesting and challenging question. For this project we will look at e-positivity and Schur positivity for certain graph classes, exploiting the relationship between graph structure and the structure of tableaux (which define Schur functions). But which graphs to consider?

A number of graph classes such as trees and cycles, have already been explored, and there has been particular focus on clawfree graphs, owing to clawfree conjectures in Stanley's original papers. In fact, a natural way to characterize graph classes is in terms of the induced subgraphs they are free of, and in graph theory, already much effort has been spent in characterizing the chromatic characteristics of graphs that are H-free, where H is some set of induced subgraphs. This literature is also at our disposal.

The key tasks and student responsibilities will be to generate examples of chromatic symmetric functions and related graphs, to explore relationships between graphs and tableaux through examples, to familiarize oneself with the proof techniques related to e-positivity and Schur positivity, to formulate conjectures, and to prove them.

The People

I consider myself lucky to have had the opportunity to have connected with an such a gifted group of international mathematicians.