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.