Equivariant index theory and noncommutative geometry

空间的数学概念（比如平面，或者我们生活的三维空间，或者甜甜圈的表面）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，而是用被称为C*-代数的某些泛化来建模。这两种数学方法之间的区别在于人们可以对“可观察对象”施加或不施加一个条件，即它们是否交换。对于量子系统，人们允许不可交换的观测。相应的C*-代数是非交换的；由于这个原因，我的学科有时被称为“非对易拓扑”（或“非对易几何”，后一个术语由该领域的主要人物之一阿兰·康恩斯推广开来，他是菲尔德奖得主）。C*代数现在广泛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。

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.