Homological invariants of manifolds and stratified spaces

摘要奖：DMS 1308306，主要研究人员：Greg Friedman主要研究人员(PI)建议使用与代数和几何拓扑学中的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。分层空间通常不是完全的流形--它们可能具有奇点--但它们是由多种层组成的。例子，包括某些群作用下的流形的代数和解析变种和商，自然地出现在许多纯数学领域和与其他科学的相互作用中。交同调是普通同调理论的一种修正，对于分层空间，它的一种形式的Poincare对偶成立。因此，这样的空间允许历史上重要的流形不变量的相交同调类似，例如签名和特征类，并提出了关于它们更广泛的背景和应用的有趣的问题。国际和平研究所提出了这一领域的几条研究路线。这包括与詹姆斯·麦克卢尔(普渡)合作，应用现代代数拓扑学的方法来研究相交(Co)链复合体的代数结构，研究有理同伦理论的相交同调版本，以及分层空间的同伦理论；与尤金妮·亨斯维尔(拉夫堡)合作，研究具有几何分析动机的签名不变量的拓扑方面；与德夫·辛哈(俄勒冈大学)合作，通过链接形式研究熊链流形的E-无穷代数的具体方面；从广义上讲，拓扑学是研究物理宇宙中的空间构形和能够模拟现实生活现象的抽象空间。例如，拓扑学家可能会研究物理弦或蛋白质链在真实的三维世界中是如何打结的，或者他或她可能会研究机器可能居住的抽象位置空间，从而允许任意数量的参数来描述各种组件的位置。首席研究人员的研究方向是“分层空间”，它同时呈现出多个维度的现象；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数量。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；尤其是拓扑学，目前正在经历应用于现实世界问题的复兴。特别是，近年来，分层空间的拓扑学在机器人运动规划、拓扑数据分析和统计生物学等应用领域以及弦理论物理等其他理论领域都有了应用。

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.