This project investigates topics at the confluence of geometric analysis, topology, and mathematical physics. The overarching goal is to develop and relate the analysis and topology of singular geometric objects. Many questions in geometry and related areas of physics, such as general relativity and holography, involve singular spaces and can be profitably understood by introducing smooth manifolds with corners, which capture the nature of singularities. Often one can recast the questions in a manner adapted to the manifold with corners and subsequently apply a toolbox of strong techniques known as "geometric microlocal analysis." One aspect of this project is to contribute to the development and application of these techniques to a large class of singular spaces (stratified spaces), allowing analytic tools to be brought to bear on algebraic and topological questions. Another aspect of the project is to promote the entry of new investigators into this active area of research. The investigator will further develop a research monograph on the subject and organize courses and conferences to disseminate these techniques and results.
An important example of a question at the confluence of geometric analysis, topology, and mathematical physics is to understand the signature of stratified spaces, such as those arising as the zeros of polynomials in several variables. The investigator and collaborators have extended the class of spaces on which the signature can be understood analytically and have shown that it satisfies some of the main properties of the signatures of smooth spaces. In this project explicit formulae will be established for the signature and its twisted analogues. The project also aims to develop a topological surgery theory for singular spaces and show that the signature operator naturally defines maps from this sequence. Other projects involve understanding an invariant of representations of the fundamental group of a locally symmetric space by changing the usual algebraic perspective to a geometric one.
