Homological invariants of manifolds and stratified spaces

流形和分层空间的同调不变量

• 批准号: 1308306
• 负责人: Greg Friedman
• 金额: $ 16.5万
• 依托单位: Texas Christian University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308306&HistoricalAwards=false
• 关键词: Homological; invariants; manifolds; stratified; spaces

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.

AbstractAward：DMS 1308306，首席研究员：格雷格·弗里德曼首席研究员（PI）建议使用与代数和几何拓扑中产生的相交同调理论和方法相关的工具来研究流形和分层空间的拓扑。 分层空间通常不完全是流形--它们可能具有奇点--但它们由流形层组成。例子，包括代数和分析的品种和流形由某些群作用的衍生物，自然发生在纯数学的许多领域和与其他科学的相互作用。交同调是普通同调理论的一个修正，它的庞加莱对偶形式在分层空间中成立。因此，这样的空间承认交叉同源类似物的历史上重要的流形不变量，如签名和特征类，并提出有趣的问题，其更广泛的背景和应用。 PI在这一领域提出了几条研究路线。 这包括与詹姆斯麦克卢尔（普渡大学）的工作，适用于现代代数拓扑的方法，研究代数结构的交叉（共）链复合体，在一个交叉同源版本的理性同伦理论，并对同伦理论的分层空间;工作与尤金Hunsicker（拉夫堡）的拓扑方面的签名不变量的动机，从几何分析;与Dev Sinha合作（俄勒冈州大学）通过链接形式研究流形上的上链的E-无限代数的具体方面;以及撰写一本关于相交同调的入门教科书。广义地说，拓扑学是对空间构型的研究，无论是在物理宇宙中，还是在可以模拟现实生活现象的抽象空间中。例如，拓扑学家可能研究物理弦或蛋白质链在真实的三维世界中如何打结，或者他或她可能研究机器可以居住的抽象位置空间，允许任意数量的参数来描述各种组件的位置。主要研究者的研究方向是“分层空间”，它同时在多个维度上表现出现象;例如，机器的运动可能会根据其当前位置表现出不同的自由度。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;特别是拓扑学，目前正在经历应用于现实世界问题的复兴。特别是，最近的应用分层空间的拓扑已经发生在这样的应用领域，如机器人运动规划，拓扑数据分析，统计生物学，以及在其他理论领域，如弦论物理。