NSF Award
1308306
Abstract<br/><br/>Award: DMS 1308306, Principal Investigator: Greg Friedman<br/><br/>The 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.<br/><br/>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.
