My research concerns dynamical systems and operator algebras and the close connections between these fields. In particular, I am interested in topological dynamical systems, particularly minimal systems on the Cantor set, systems arising from aperiodic tilings, symbolic dynamical systems and the hyperbolic systems associated with Smale's program for Axiom A dynamics. In operator algebras, I am mainly interested in C*-algebras, groupoid C*-algebras, K-theory for C*-algebras and other aspects of Alain Connes' program in noncommutative geometry.

From a topological dynamical system, there are several constructions of C*-algebras. This provides a fascinating link between the two subjects. On the one hand, it produces interesting examples of C*-algebras. Some, such as the Cuntz-Krieger algebras constructed from subshifts of finite type and the C*-algebras associated with irrational rotations of the circle, have had a fundamental impact on the subject. It is insightful to see how dynamical properties manifest themselves in the C*-algebras. On the other hand, techniques which are used in the study of C*-algebras can often give new information about the dynamics. As a specific example, starting from a dynamical system, constructing the C*-algebra and then taking its K-theory provides an interesting invariant, rather like homology of spaces. In my work on minimal dynamical systems on Cantor sets, I (and my co-workers) used K-theory to obtain a complete invariant for orbit equivalence for many systems. In addition, our work brought many C*-algebra ideas (such as Bratteli diagrams) to effective use in dynamics.

Interests

• Operator algebras
• C*-algebras
• Topological dynamics
• Symbolic dynamics
• K-theory

Courses

• Fall 2024:
• Spring 2025: 听
• Summer 2025:

Current Projects

Selected Publications

• With R.J. Deeley and K.R. Strung, Constructing minimal homeomorphisms of point-like spaces and a dynamical presentation of the Jiang-Su algebra, to appear, Journal fur die Reine und Angewandte Mathematik
• With A. Julien, Spectral triples for subshifts, preprint.
• With N.D. Burke, Markov partitions and homology for n/m-solenoids, to appear, Ergod. Th. and Dynam. Sys.
• With J. Kaminker and M.F. Whittaker, K-theoretic duality for hyperbolic dynamical systems, to appear, Journal fur die Reine und Angewandte Mathematik
• A homology theory for Smale spaces, Memoirs A.M.S.
• With T. Giordano, H. Matui and C.F. Skau, Orbit equivalence for Cantor minimal Z^d-systems, Invent. Math., 179(2010), 119-158.
• With T. Giordano, H. Matui and C.F. Skau, Orbit equivalence for Cantor minimal Z^2-actions, Journal A.M.S., 21(2008), 863-892.
• With J. Anderson, Topological invariants for substitution tilings and their C*-algebras , Ergodic Th. and Dynam. Sys. 18(1998), 509-537.
• With T. Giordano and C.F. Skau, Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469(1995), 51-111.
• With R.H. Herman and C.F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics , International J. Math. 6(1992), 827-864.