Professor  Ph.D., University of Toronto  Specializations:  Harmonic and functional analysis  Current research interests Topological groups and flows, and  associated function and operator algebras    Publications Homepage Conference Photographs

My research is centred around topological groups, compact right topological groups, flows and C*-algebras. These mathematical objects are both algebraic and topological in nature, are of great interest to mathematicians, and are widely studied by them; they are also very useful to physicists, statisticians and social scientists. My study of these concepts uses powerful tools from harmonic analysis, topological dynamics and functional analysis.

One area of my work has its origins at the beginning of this century, in the work of Harald Bohr on almost periodic functions on the real line. Since then the subject has grown enormously and now includes the study of the algebras of weakly almost periodic functions, almost automorphic functions, distal functions and many other functions of "almost periodic type" on groups and semigroups G. As well as the tools mentioned above, a unifying concept in this work is the appropriate notion of compactification of G, which is like the Stone-Cech compactification, except that account is also taken of the algebraic structure of G. The structure of the relevant compactifications plays an important role in determining functional analytic and dynamical properties of the algebras and of G.

In other work, I study C*-algebras generated from operator equations (analogous to UV = \lambda VU, the equation generating the much-studied "irrational rotation" C*-algebras) and the connection of these algebras with some special groups and flows. I also study the representation theory of compact right topological groups; the results achieved indicate that this theory will be difficult, with much work still to be done. An important and interesting aspect of my work is the study of examples, structure and other properties of flows and compact right topological groups. A notable success in this area was the discovery of Haar measure on compact right topological groups - that is, a probability measure on the group that is both left and right invariant, and unique as such; this discovery was made in joint work with coauthor J.S. Pym.

Telephone: 519-661-3638
Fax: 519-661-3610

Email: milnes at uwo.ca