Grant: Discovery Grants Program - Individual

My ongoing research is centred around topological groups, compact right topological groups, flows and C*-algebras. These mathematical concepts are widely studied by mathematicians; 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 many algebras 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 taken also 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 ongoing work, I study C*-algebras (closed algebras of operators on Hilbert space) 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; there are many interesting unfinished aspects of this work.    A further important 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.