Research

The following is a list of my available research works with shortened abstracts:

• The K-theory of the C*-algebras associated to a rational function, arxiv:2307.13420 (submitted).

  Abstract: We compute the K-theory of the three C*-algebras associated to a rational function, thought of as a dynamical system acting on its Julia set, Fatou set, or the entire Riemann sphere. Our results yield new dynamical invariants for rational functions and a C*-algebraic formulation of the Density of Hyperbolicity Conjecture for quadratic polynomials.

• KK-duality for self-similar groupoid actions on graphs, arxiv:2302.03989 (submitted). Joint with Nathan Brownlowe, Alcides Buss, Daniel Gonçalves, Aidan Sims, and Michael F. Whittaker

  Abstract: We prove that two naturally associated C*-algebras to a regular and contracting self-similar groupoid are Spanier-Whitehead dual (in KK-theory) to each other by showing they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from the self-similar limit space.

• Renormalization procedures for C*-algebras (MSc. Thesis at the University of Victoria), (http://hdl.handle.net/1828/13285). Supervised by Ian F. Putnam.

  Abstract: We introduce renormalization procedures for groupoids and C*-algebras, in analogy to renormalization procedures for families of dynamical systems. We prove a C*-analog to Masur’s unique ergodicity criterion for flat surfaces and apply this criterion to show a variety of C*-algebras have unique trace.