The broad goal of the PI's research is to study infinite dimensional objects arising in geometry and topology using the tools of homotopy theory. The PI is currently interested in infinite dimensional transformation groups like Loop groups and Symplectomorphism groups. The topology of these groups may be studied via their proper actions on certain associated spaces. For example, the Symplectormorphism group of a Symplectic manifold may be studied via its action on the space of compatible complex structures. A careful analysis of this space allows us to compute various homological invariants of the Symplectomorphism group, like homology and homotopy. Similarly, the Loop group may be studied via its action on the space of connections on a trivial bundle. The orbit structure of these spaces allows one to decompose the homotopy type of the infinite dimensional group in terms of compact Lie groups. There are numerous applications of this decomposition. It may be exploited to understand the topology of other infinite dimensional objects that admit the Loop group or the Symplectomorphism group as their structure group. An example of such an object is the Loop space of a manifold.
In recent years, Lie groups have become increasingly important in mathematics and physics. They may be seen as groups of symmetries of various geometric objects. While finite dimensional Lie groups are well understood, the infinite dimensional ones are just beginning to be understood. Due to their infinite nature, their study is a difficult task. I have been successful in developing methods to systematically decompose these infinite dimensional Lie groups into finite dimensional compact Lie groups, hence making it possible to study them using standard techniques of geometry and topology. My current research involves extending these decomposition methods to other infinite dimensional transformation groups.
