Equivariant index theory and noncommutative geometry

等变指数理论和非交换几何

• 批准号: 327638-2011
• 负责人: Emerson, Heath
• 金额: $ 1.09万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572081
• 关键词: Equivariant; index; theory; noncommutative; geometry

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.

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，即它们是否可交换。 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。 在拓扑学中，人们有空间的“上同调群”的概念。它们是空间的不变量。类似地，每个C*-代数都可以与它的K-理论相关联：一个阿贝尔群。我的主要兴趣是在K-理论和指数理论和研究推广和改进的某些经典不变量的空间（如欧拉特征）使用C* 代数和K-理论技术。例如，经典拓扑学（空间研究）中最重要的不变量之一是空间对称（自映射）的莱夫谢茨数，定义为上同调诱导映射的迹（如矩阵的迹）。莱夫谢茨不动点定理是经典拓扑学的基石，它说这个数目大致上等于地图的不动点数目。该定理等同于两个非常不同类型的不变量，因此它的重要性。最近我发现了一些不同的类似物和推广的莱夫谢茨不动点公式涉及这个新的“量子”（或“非交换”）拓扑。这些结果使用Kasparov理论的技术。我的建议是继续研究这些非交换莱夫谢茨不动点定理，以及非交换拓扑学中的其他兴趣点。