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】本研究包括两个不同的项目；离散群的诺维科夫猜想和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。第一个是由于观察到，诺维科夫猜想和精确性猜想所适用的群的种类在很大程度上是一致的；例如，这两个类都包含可调群、双曲群和考克斯特群。这个项目是与J. Kaminker合作的。第二种是基于Berezin-Toeplitz量化可以用dirac型算子族的谱性质来分析的思想。这个项目与j·特劳特联合。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。精确性猜想是纯解析的。这两个来自完全不同数学分支的猜想之间可能存在联系，这多少有些令人惊讶，但却得到了经验证据的支持。我们计划更充分地发展这些猜想之间的关系。这一研究涉及到包括鲍姆-康恩斯猜想在内的许多重要的突出问题。数学和物理学之间联系的核心是量子力学理论。本文提出了一种基于微分方程组的几何性质来分析量子力学系统的新方法。这项工作将有助于从数学物理角度更好地理解许多量子力学系统。