# Ian Putnam

##### Position

##### Credentials

Ph.D., California, Berkeley

##### Contact

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 2022:
- Spring 2023: MATH 465: Topics in Topology MATH 493: Topics in Pure Mathematics MATH 531: Functional Analysis
- Summer 2023:

### 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.