Homological invariants of manifolds and stratified spaces

项目编号：DMS 1308306，项目负责人：Greg friedman。项目负责人（PI）提出利用代数和几何拓扑中的交同调理论和方法研究流形和分层空间的拓扑结构。分层空间通常不是流形——它们可能具有奇点——但它们是由流形层组成的。在纯数学的许多领域以及与其他科学的相互作用中，自然会出现一些例子，包括由某些群作用产生的代数和解析变量以及流形商。交同调是普通同调理论的一种修正，它在分层空间中具有庞加莱对偶的一种形式。因此，这样的空间承认历史上重要的流形不变量（如签名和特征类）的交同源类似物，并提出了关于其更广泛的背景和应用的有趣问题。PI在这一领域提出了几个研究方向。这包括与James McClure （Purdue）合作，将现代代数拓扑方法应用于交（co）链配合物的代数结构、交同调版本的理性同伦理论和分层空间的同伦理论的研究；与Eugenie Hunsicker （Loughborough）合作，研究几何分析中具有动机的特征不变量的拓扑方面；与Dev Sinha（俄勒冈大学）合作，通过连接形式研究流形上的e -∞协链代数的具体方面；并编写了交同调入门教材。从广义上讲，拓扑学是对空间结构的研究，既包括物理宇宙，也包括可以模拟现实现象的抽象空间。例如，拓扑学家可能会研究现实三维世界中的物理弦或蛋白质链是如何打结的，或者他或她可能会研究机器可能占据的位置的抽象空间，允许任意数量的参数来描述各种组件的位置。首席研究员的研究路线涉及“分层空间”，同时在多个维度上展示现象；例如，一台机器的运动可能会根据其当前位置表现出不同数量的自由度。虽然这些研究项目往往是纯理论的，但随着时间的推移，理论结果会渗透到应用中；特别是拓扑，目前正在经历应用程序解决实际问题的复兴。特别是，最近分层空间拓扑的应用已经出现在机器人运动规划、拓扑数据分析和统计生物学等应用领域，以及其他理论领域，如弦理论物理。

AbstractAward: DMS 1308306, Principal Investigator: Greg FriedmanThe Principal Investigator (PI) proposes to study the topology of manifolds and stratified spaces using tools related to intersection homology theory and methods arising in algebraic and geometric topology. Stratified spaces are usually not quite manifolds - they may possess singularities - but they are composed of manifold strata. Examples, including algebraic and analytic varieties and quotients of manifolds by certain group actions, occur naturally in numerous fields of pure mathematics and in interactions with other sciences. Intersection homology is a modification of ordinary homology theory for which a form of Poincare duality holds for stratified spaces. Consequently, such spaces admit intersection homology analogues of historically important manifold invariants, such as signatures and characteristic classes, and raise interesting questions regarding their broader context and applications. The PI proposes several lines of research in this area. This includes work with James McClure (Purdue) to apply methods of modern algebraic topology to research on the algebraic structures of intersection (co)chain complexes, on an intersection homology version of rational homotopy theory, and on homotopy theory of stratified spaces; work with Eugenie Hunsicker (Loughborough) on topological aspects of signature invariants with motivations from geometric analysis; work with Dev Sinha (University of Oregon) to study concrete aspects of the E-infinity algebra of cochains on manifolds via linking forms; and the writing of an introductory textbook on intersection homology.Broadly speaking, topology is the study of spatial configuration, both in the physical universe and of abstract spaces that can model real-life phenomena. For example, a topologist might study how a physical string or protein strand is knotted in the real three-dimensional world, or he or she might study the abstract space of positions that a machine could inhabit, allowing for an arbitrary number of parameters that describe the positions of various components. The principal investigator's line of research concerns "stratified spaces" that simultaneously exhibit phenomena in a multitude of dimensions; for example, a machine's motions might exhibit different numbers of degrees of freedom depending upon its current position. While these research projects tend to be purely theoretical, theoretical results percolate over time into applications; topology, in particular, is currently experiencing a renaissance of applications to real-world problems. In particular, recent applications of the topology of stratified spaces have occurred in such applied fields as robot motion planning, topological data analysis, and statistical biology, as well as in other theoretical fields, such as string theory physics.