K-theory, Dynamics, and Intersection

K 理论、动力学和交集

• 批准号: 1104355
• 负责人: John Klein
• 金额: $ 13.32万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104355&HistoricalAwards=false
• 关键词: theory; Dynamics; Intersection

The PI proposes two projects that interface algebraic and differential topology and K-theory. The first of these aims to strengthen the relationship between torsion invariants and the counting of closed orbits in dynamical systems. The goal here is to generalize Milnor's equation relating Reidemeister torsion to the Lefschetz zeta function. One aspect of the project will lead to a version of this equation which holds for families of dynamical systems that will relate a higher K-theory invariant to an invariant counting periodic orbits. The main tools will come from Waldhausen's work on algebraic K-theory and the functor TR, which is the topologists' version of the topological de Rham-Witt complex. The second project will study multi-relative version of intersection theory which will give insights beyond the metastable range. We will study obstructions to deforming a map from a manifold into another one off of a finite collection of pair-wise disjoint submanifolds, assuming that the map can be deformed off of the union of any proper sub-collection. The obstruction will live in a direct sum of bordism groups. In a certain range, the obstruction will capture the entire story. Applications of this theory will be given in embedding theory and also in the theory of link homotopy.The proposed research is largely about "manifolds" which are topological spaces that satisfy a certain homogeneity property. Locally speaking, all manifolds are alike in that at any point one sees a copy of Euclidean space. It is the global structure of manifolds that makes them interesting objects of study. Typically, algebraic topologists study manifolds by assigning certain algebraic quantities, called "invariants," to them, which measure global topological structure. Manifolds having different invariants can then be distinguished from one another. Manifolds arise naturally in physics, chemistry and biology as spaces of solutions of a suitably "nice" set of algebraic equations modeling the scientific object of study (space-time, atoms, dynamical systems, etc.) . Manifolds play a central role in mathematics. It is often the case that mathematical questions about manifolds can be formulated in terms of parametrized families of functions between spaces associated with manifolds. Homotopy theory is a subject designed to tackle questions about such families of functions. K-theory is an algebraic theory which is a receptacle for invariants of manifolds. The PI will research certain kinds of manifold questions which can be analyzed using homotopy theory and K-theory.

PI提出了两个方案，将代数和微分拓扑学与K-理论相结合。第一个目的是加强动力学系统中挠率不变量与闭合轨道计数之间的关系。这里的目的是将米尔诺方程推广到Lefschetz Zeta函数，该方程将Reidemister挠率与Lefschetz Zeta函数联系起来。该项目的一个方面将导致这个方程的一个版本，它适用于动力学系统族，它将把更高的K-理论不变量与不变计数周期轨道联系起来。主要工具将来自Waldhausen在代数K理论和函子tr方面的工作，函子tr是拓扑学家版本的拓扑德罗姆-维特复形。第二个项目将研究多相对版本的交集理论，这将给出亚稳态范围之外的见解。我们将研究从一个流形到有限个成对不相交子流形集合的映射变形的障碍，假设映射可以从任何真子集合的并集变形出来。障碍将直接存在于博尔德主义团体的总和。在一定范围内，障碍物将捕捉到整个故事。这一理论将在嵌入理论和链同伦理论中得到应用。所提出的研究主要是关于流形，即满足一定齐性性质的拓扑空间。局部地讲，所有流形都是相似的，因为在任何一点都可以看到欧几里得空间的副本。正是流形的全球结构使它们成为有趣的研究对象。通常，代数拓扑学家通过给流形分配某些代数量来研究流形，这些代数量被称为“不变量”，用来测量全局拓扑结构。然后，具有不同不变量的流形可以彼此区分。流形自然而然地出现在物理、化学和生物学中，作为一组适当的、对科学研究对象(空间-时间、原子、动力系统等)建模的代数方程的解的空间。流形在数学中起着核心作用。通常情况下，关于流形的数学问题可以用与流形相关的空间之间的函数的参数化族来表示。同伦论是一门旨在解决关于这类函数族的问题的学科。K-理论是一种接受流形不变量的代数理论。PI将研究某些类型的流形问题，这些问题可以用同伦理论和K理论来分析。