My recent research is mainly concerned with gauge theory, in particular a finite dimensional approximation of the Seiberg-Witten equation obtained by M. Furuta. Using this and equivariant homotopy theory, he showed some remarkable results including the so-called 10/8-inequaltity. In joint work with him and others we have improved some of his results. I hope recent developments in elliptic cohomology lead to a new invariant in our setting.
We are now attempting to apply this theory to transformation group theory in 4-dimensional manifolds. I would like to understand equivariant homotopy theory through various gauge theoretic phenomena.