Chromatic Symmetric Functions: Solving Algebraic Conjectures Using Graph Theory

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Many processes depend largely on the innate symmetries of natural objects, and how the movement of these objects in space affects them; for example, determining the energy states of a subatomic particle and predicting transitions between them, or distinguishing communication signals from random noise. One way to study such objects is to perturb them by rotations and analyze the resulting interactions. Because of symmetry, some rotations will produce an identical result to others. These symmetries may be encoded by symmetric functions, multivariate polynomials that are fixed under any permutation of the variables. The study of different objects can then be represented by considering symmetric functions within the context of a given algebraic structure or theory. As examples, for subatomic particles and quantum physics we consider the Fock space, and for signal identification we consider random matrix theory. The long-term goal of this research is to study symmetric functions through the theory of graphs. Graphs arise frequently in research to model discrete systems and their relationships. Graph theory is one of the most studied areas of discrete mathematics, and the main approach to this research will be to interpret symmetric functions in terms of graphs and their structure to apply a much broader range of knowledge. A starting point will be the chromatic symmetric function X_G, currently the only major connection between graphs and symmetric functions. This program will expand our knowledge helping us to understand the information X_G encodes about graphs, and to generalize X_G to a stronger form based on K-theory and other ideas from modern algebra. Additionally, this program will create graph-based constructions to represent algebraic objects such as LLT polynomials and plethysms of Schur functions, thus providing new avenues of attack for notoriously difficult problems in algebraic combinatorics such as interpreting Macdonald polynomials, and determining branching rules for representations of symmetric groups. The long-term vision of this research anticipates that these two paths will converge and build upon each other: chromatic symmetric functions are already related in special cases to LLT polynomials, which in turn are used in general symmetric function theory to study the movement of particles between quantum states in quantum mechanics. Thus, this research aims to form a bridge connecting two well-studied areas of mathematics by building from both sides. Such a bridge will greatly advance results and approaches in both fields, as techniques from one may be easily applied to the other. Students taking part in this program will learn valuable skills at the forefront of two major areas of discrete mathematics with immediate applications to positions in academia or the computer science industry.