William Dugan (M.Math.) and Tom Kelly (Ph.D.) will be recognized as University Finalists for the Governor General's Gold Medal at the Spring 2020
William wrote an M.Math. thesis on "Sequences of trees and higher-order renormalization group equations" under the supervision of Professor Karen Yeats. His work utilized ideas from mathematical physics, combinatorics, and algebra. Here is the abstract of his thesis:
In 1998, Connes and Kreimer introduced a combinatorial Hopf algebra HCK on the vector space of forests of rooted trees that precisely explains the phenomenon of renormalization in quantum field theory. This Hopf algebra has been of great interest since its inception, as it connects the disciplines of algebra, combinatorics, and physics, providing interesting questions in each. In this thesis we introduce the notion of higher-order renormalization group equations, which generalize the usual renormalization group equation of quantum field theory, and further define a corresponding notion of order on certain sequences of trees constituting elements of the completion of HCK. We also give an explication of a result, due to Foissy, that characterizes which sequences of linear combinations of trees with one generator in each degree generate Hopf subalgebras of HCK.
William is currently pursuing Ph.D. studies at the University of Massachusetts, Amherst.
Tom Kelly, who was recently recognized as winner of the Faculty of Mathematics Doctoral Prize competition, was supervised by Professor Luke Postle. His thesis, Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm, made significant contributions to long-standing open problems in graph colorings. Tom is currently a postdoctoral fellow at the University of Birmingham (UK).