The Novikov and Exactness Conjectures for Discrete Groups

离散群的诺维科夫猜想和精确性猜想

• 批准号: 0071402
• 负责人: Erik Guentner
• 金额: $ 6万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071402&HistoricalAwards=false
• 关键词: Novikov; Exactness; Conjectures; Discrete; Groups

AbstractGuentnerThe proposed research comprises two distinct projects; therelationship between the Novikov and exactness conjectures fordiscrete groups, and the use of global analytic techniques to studyquantum mechanical systems. The first is motivated by the observationthat the classes of groups for which the Novikov and exactnessconjectures are known coincide to a large degree; both classes containamenable groups, hyperbolic groups and Coxeter groups, for example.This project is joint with J. Kaminker. The second is based on theidea that the Berezin-Toeplitz quantization can be analyzed usingspectral properties of family of Dirac-type operators. This projectis joint with J. Trout.The Novikov conjecture, one of the most important problems intopology, has stimulated a tremendous amount of mathematical researchover the last thirty years. The exactness conjecture is purelyanalytic. That there could exist a connection between these twoconjectures from entirely different branches of mathematics issomewhat surprising, but nevertheless is supported by empiricalevidence. We plan to develop more fully the relationship betweenthese conjectures. This research has bearing on a number of importantoutstanding problems including the Baum-Connes Conjecture. At theheart of the connection between mathematics and physics is the theoryof quantum mechanics. We propose a new method to analyze quantummechanical systems based on geometric properties of certain systems ofdifferential equations. This work should lead to a betterunderstanding of a number of quantum mechanical systems frommathematical physics.

摘要Guentner 拟议的研究包括两个不同的项目：诺维科夫猜想和离散群精确猜想之间的关系，以及使用全局分析技术来研究量子力学系统。 第一个是由这样的观察所激发的：已知诺维科夫猜想和精确性猜想的群类在很大程度上是一致的；例如，这两个类都包含可修改的群、双曲群和 Coxeter 群。该项目是与 J. Kaminker 联合进行的。 第二个基于这样的想法：可以使用狄拉克型算子族的谱特性来分析 Berezin-Toeplitz 量化。 该项目与 J. Trout 联合开展。诺维科夫猜想是拓扑学中最重要的问题之一，在过去的三十年里激发了大量的数学研究。 精确性猜想纯粹是分析性的。 这两个来自完全不同的数学分支的猜想之间可能存在联系，这有点令人惊讶，但仍然得到了经验证据的支持。 我们计划更充分地研究这些猜想之间的关系。 这项研究涉及包括鲍姆-康尼斯猜想在内的许多重要的悬而未决的问题。 数学和物理学联系的核心是量子力学理论。 我们提出了一种基于某些微分方程组的几何性质来分析量子力学系统的新方法。 这项工作应该可以使我们更好地理解数学物理学中的许多量子力学系统。