Equivariant index theory and noncommutative geometry

在量子力学之前，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)历史上一直被用来对物理系统进行建模。量子系统更好的建模方法不是空间，而是它们的某些推广，称为C*-代数。这两种数学方法之间的不同之处在于一个条件，即它们是否通勤，是不是可以强加给“可观察者”。对于量子系统，人们允许非对易可见。相应的C*-代数是非交换的；出于这个原因，我的学科有时被称为‘非交换拓扑’(或‘非交换几何’，后者被该领域的主要人物之一、菲尔德奖牌获得者阿兰·康内斯推广)。C*-代数现在被广泛地应用于许多不同的数学领域。最近，它们也被用于弦理论和物理学中。

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.