Principal Investigator: Nitya Kitchloo
This proposal emphasizes three concrete projects. The aim in the first project is to give a topological interpretation to the class of highest weight representations of infinite dimensional Kac-Moody groups. The PI plans to construct a variant of equivariant K-theory on the category of spaces that admit a proper action of a Kac-Moody group. In particular, one hopes to relate the class of highest weight representations with elements in this K-theory. This project will generalize recent work of Freed-Hopkins-Teleman. The authors cited gave a topological interpretation of the positive energy representations of Loop groups. This is a special case of an affine Kac-Moody group. This research program will study more general Kac-Moody groups. The goal of the next project is to understand the topology of the space of compatible complex structures on a symplectic 4-manifold. This builds on an earlier joint project of mine where we showed that for rational ruled surfaces with arbitrary symplectic forms, the space of compatible complex structures is weakly contractible. In that project, we use a comparison technique that relates the space of complex structures with the space of almost complex structures, and decompose the latter using the theory of J-holomorphic curves. In the proposed project, we will attempt to apply our techniques to other 4-dimensional symplectic manifolds. As a by product of our techniques, we hope to derive consequences about the group of symplectomorphisms of 4-dimensional manifolds. In the final project, we study a new cohomology theory called real Johnson-Wilson theory, and develop computational tools to make it accessible. Real Johnson-Wilson theory is constructed by taking fixed points with respect to an involution on standard Johnson-Wilson theory. These are 2-local theories indexed over the integers, with the first theory being 2-localized real K-theory. We plan to use our tools, along with techniques in Algebraic Topology, to settle various old questions regarding immersions of real projective spaces in euclidean space. Previously, the best know results used the theory TMF (Topological Modular forms).
The general focus of my projects is to explore the interaction between geometry and topology. The objects of study in geometry are fundamental physical objects (in possibly more than three dimensions), that are endowed with various structure. The structure one associates to these spaces could be as crude as the homotopy type (the information about the space that is preserved under all continuous deformations), or one may be interested in more subtle structure like the differential structure (the local information about the space that makes it smooth). One may refine the smooth structure even further and study symplectic, or complex structure. These questions are motivated by fundamental inquiries into our physical universe. Indeed, mathematical physics is a rich source of questions and ideas. Given a particular structure of interest, one may ask the question of how to enumerate them. In other words, one is interested in ways to parametrize all possible structures on a space. For example, John Milnor was the first person to really show that spheres admit multiple non-equivalent smooth structures, starting with the 7-dimensional sphere which admits 28 such structures. One of my projects is to study the question that inquires into the number of complex structures in 4-dimensional smooth spaces. The techniques that one uses to understand the space of complex structures is to decompose it into pieces that have some significance in terms of their symmetry. Then one studies these pieces using the invariants of topology and then reassembles these pieces to get a coherent picture on the global level. The question of symmetry is also important in other projects that I propose. For example, the linear symmetries of a vector space (like euclidean space), which preserve some property, are known as representations. Another project aims to study a class of such representations that act on infinite dimensional vector spaces. We will try to encode these objects via a topological invariant known as equivariant K-theory. Such invariants are generally known as cohomology theories and form natural receptacles for algebraic information derived from geometry. In a third and final project, the PI plans to develop such a cohomology theory, called real Johnson-Wilson theory. We hope to justify its importance by showing that it helps to solve an old geometric question about the minimum dimension of euclidean space where a certain class of spaces, known as real projective spaces, may be realized.
