Richard M. Kane Professor Ph.D., University of Waterloo (1973) Current research interests Homology of classifying spaces of Lie groups and finite loop spaces, invariant theory of (pseudo)reflection groups Publications

Telephone: 519-661-3639
Fax: 519-661-3610

E-mail: rkane@uwo.ca

My research studies various types of multiplicative objects. Compact Lie groups, finite loop spaces and finite Hopf spaces represent increasing levels of abstraction. They are geometric objects which codify certain basic algebraic operations, starting from matrix multiplication. I have research projects connected with each of these levels, principally concerning homology theory. A homology theory is, in essence, an algorithm for converting spaces into algebra. A pervasive problem for all of the above types of multiplicative objects is understanding the torsion in their homology.

The "Topology of Lie Groups" is a longstanding program with the goal of effectively understanding how the data of Lie groups (which makes them so important to Science - e.g. Weyl groups or representations) is reflected in structures of algebraic topology such as homology.

Torsion in homology and an explanation of homology via generalized invariants are two interconnected problems that now seem capable of considerable progress. Finite loop spaces were extensively studied in the 1970's and 1980's. They are generalizations of Lie groups, and the main question about them is the extent to which they and their homology satisfy structure theorems analogous to ones obtained for Lie groups. Much has been obtained, but their torsion in particular remains unstudied. In the case of Hopf spaces a great deal is known about their homology torsion but the algebraic patterns satisfied by the homology of Hopf spaces still remains elusive.