The Novikov and Exactness Conjectures for Discrete Groups

摘要Guentner提出的研究包括两个不同的项目：离散群的Novikov和精确度之间的关系，以及使用全局分析技术来研究量子力学系统。 第一个动机是观察到的类群的诺维科夫和exactnessitures是已知的在很大程度上重合;这两个类containable群，双曲群和Coxeter群，例如。 第二种是基于Berezin-Toeplitz量子化可以用Dirac-type算子族的谱性质来分析的思想。 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，它激发了大量的数学研究。 精确性猜想是纯分析的。 这两个完全不同的数学分支之间可能存在联系，这有点令人惊讶，但无论如何都得到了数学证据的支持。 我们计划更充分地发展这些组织之间的关系。 这项研究关系到包括Baum-Connes猜想在内的一些重要的悬而未决的问题。 数学和物理之间联系的核心是量子力学理论。 我们提出了一种新的方法来分析量子力学系统的基础上的几何性质的某些系统的微分方程。 这项工作将使我们从数学物理学中更好地理解一些量子力学系统。

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.