My research.

The area of my research is algebraic and geometric topology. These fields are concerned with the mathematical theory of shape recognition. Certain (theoretical) machines, called cohomology theories, have proven particularly powerful for studying shapes (called spaces by topologists). Most of my research is concerned with relations between cohomology theories and supersymmetry, a concept from theoretical physics that has its origins in quantum field theory (QFT) and string theory. A very pretty example of the interaction between algebraic topology and supersymmetry is the construction of the de Rham complex, the basic geometric tool that leads to simplest example of a cohomology theory, using the natural operation of the super point on the parity changed tangent bundle of a manifold. This construction is well-known to super geometers, but since I could not find a precise reference, I wrote up detailed account, which can be found here. The result may be interpreted in terms of (0|1)-dimensional supersymmetric QFTs. This is described in [HKST].

It turns out that the relation between supersymmetry and de Rham cohomology is only the simplest case in a series of results relating supersymmetric QFTs and cohomology theories. About five years ago, Stephan Stolz and Peter Teichner explained how supersymmetry comes up in the context of K-theory and (unfortunately still conjecturally) elliptic cohomology, see [ST1]. The role and implementation of supersymmetry in their work was the main topic of my Ph.D. thesis [H1], which I wrote under the direction of Peter Teichner at the University of California at San Diego and Berkeley. The first half deals with the relation between K-theory and Euclidian field theories. A part of this is also contained in the joint paper [HST] with Stephan Stolz and Peter Teichner. The second half of my thesis concerns the relation between supersymmetric conformal field theories and integral modular forms (and eventually elliptic cohomology TMF, we hope). The proposed notion of supersymmetry is different from the one developed by Stolz and Teichner, see [ST2], but is closely related. In any case, my thesis contains a proof that their approach (as well as mine) satisfies the conditions they asked for in [ST1]. I hope that my slightly different approach will prove useful, because it connects the work of Stolz and Teichner to SUSY curves, the much studied analogues of Riemann surfaces in complex super geometry.

The paper [H2] is the write-up of the talk I gave at the Talbot workshop on topological modular forms in the Spring of 2007. It describes one of the basic steps in the homotopy-theoretic construction of TMF following Hopkins and Miller.

Finally, [H0] is my Diploma thesis. It is essentially an application of the surgery theory developed by M. Kreck to a classification question concerning 6-dimensional spaces (manifolds).