Algebraic topology and its applications. Algebraic topology is a very effective tool to study the global properties of geometric objects. For example, take the surface of a ball and divide it into triangles; now count the number of faces, add the number of vertices and subtract the number of edges; no matter how you choose your triangles, the result will always be 2. Do the same with the surface of a donut and the result is always 0. These numbers were already known by Euler and are foreshadows of homology developed in the 20th century. By now the basic ideas of algebraic topology have permeated nearly every branch of research in mathematics.
My own research has been motivated by questions in quantum physics and string theory. In particular, I have contributed to our understanding of the 'space of surfaces'.