Equivariant index theory and noncommutative geometry

空间的数学概念（如平面，或我们生活的三维空间，或甜甜圈的表面）在历史上被用来模拟物理系统，直到量子力学。量子系统最好不是用空间来建模，而是用它们的某些推广来建模，称为C*-代数。这两种数学方法之间的区别在于可以对“可观测物”施加或不施加一个条件，即它们是否对易。 对于量子系统，允许非对易的可观测量。相应的C*-代数是非交换的;由于这个原因，我的课题有时被称为“非交换拓扑”（或“非交换几何”，后一个术语已由该领域的主要人物之一、菲尔德奖章获得者阿兰·康纳斯推广。）C*-代数现在被广泛应用于许多不同的数学领域。最近，它们也被用于弦论和物理学。

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.