In this proposal we propose several projects of geometric significance. Among the projects proposed is the study of the stable sympelctic category. This category is the stabilization of A. Weinstein's symplectic category (which is a domain category for geometric quantization). We achieve stabilization by inverting the symplectic manifold given by the complex plane. The resulting stable category is extremely tractable, and retains a lot of structure (like an action of the Grothendieck-Teichmuller group that was conjectured by Kontsevich). Another project proposed is the continuing study of real Johnson-Wilson theories, with collaborator W.S.Wilson. This is a computationally accessible theory that contains higher-level information which can be used to solve several geometric problems like the non-immersion problem. Among other projects proposed include the study of the homotopical and representation theoretic properties of Kac-Moody groups. These groups are of interest in mathematical physics and topology.
The major thrust of the above proposal is to construct and apply (homotopical) techniques and methods designed to better understand hard geometric questions that may be of interest to a wide variety of mathematicians and mathematical-physicists. One such technique is called stabilization, which suitably linearizes a possibly intractable question, thereby making it accessible. The process often allows us to gain a lot of insight into the original problem. Another technique studied is the study via classifying spaces. The classifying space is a construction that takes the set of symmetries of a geometric object, and encodes them into a space that can then be studied in standard ways to recover information about the original geometric object.
