Equivariant index theory and noncommutative geometry

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The mathematical notion of a space (like the plane, or the three-dimensional space we live in, or the surface of a donut) was historically used to model physical systems, until quantum mechanics. Quantum systems are better modeled not by spaces but by certain generalizations of them called C*-algebras. The difference between the two mathematical approaches lies in a condition one may impose or not impose on `observables', namely whether or not they commute. For quantum systems, one allows noncommuting observables. The corresponding C*-algebras are noncommutative; my subject, for this reason, is sometimes called `noncommutative topology' (or `noncommutative geometry,' the latter term has been popularized by one of the main figures in the field, Alain Connes, a Field's medalist.) C*-algebras are now used extensively in many different fields of mathematics. Recently they have been used in string theory, in physics, as well. In topology, one has the notion of the `cohomology groups' of a space. These are invariants of the space. Similarly, one can associate to every C*-algebra its K-theory: an abelian group. My main interest is in K-theory and index theory and in studying generalizations and refinements of certain classical invariants of spaces (like the Euler characteristic) using C*-algebra and K-theory techniques. For example, one of the most important invariants in classical topology (the study of spaces) is the Lefschetz number of a symmetry (self-map) of the space, defined as the trace (as in the trace of a matrix) of the induced map on cohomology. The Lefschetz fixed-point theorem, a cornerstone of classical topology, says that this number is equal, roughly speaking, to the number of fixed-points of the map. The theorem equates two very different types of invariants, whence its importance. Recently I have discovered a number of different analogues and generalizations of the Lefschetz fixed-point formula involving this new `quantum' (or `noncommutative') topology. These results use the technology of Kasparov theory. My proposal is to continue the study of some of these noncommutative Lefschetz fixed-point theorems, and other points of interest in noncommutative topology.