International conference
Combinatorial Methods in Physics and Knot Theory

The problem of studying the intersection theory on Hurwitz spaces naturally arises in connection with the Hurwitz problem. The latter consists in enumeration of ramified coverings of the $2$-sphere, with prescribed ramification data. It is also closely related to the computation of Gromov--Witten invariants --- the invariants of compact complex manifolds counting maps of complex curves to the manifold. The study of the intersection theory on Hurwitz spaces has been initiated in [1,4].
We discuss a new approach to the intersection theory on Hurwitz spaces. This approach is based on the theory of universal polynomials originating in the work by R. Thom in early 60ies. It concerns general holomorphic mappings $f:M\to N$ of compact complex manifolds (which we assume, for definiteness, being of the same dimension). The main theorems of the theory postulate, in various situations, the existence of universal polynomials in the relative Chern classes of $f$, expressing the cohomology classes of the closures of loci of points in $M$ where $f$ acquires given singularities. This theory was recently extended to the case of multisingularities by M. Kazarian [2] whose results allow one to describe as universal polynomials in the pushforwards of the relative Chern classes the cohomology classes in $N$ represented by the loci of points with prescribed singularities at the preimages.
We apply the theories of universal polynomials (and their further extensions) to the Hurwitz spaces treated as families of meromorphic functions on complex curves [3]. Since the domain of definition of a function is one dimensional, the classification of singularities is relatively simple, and there is a hope for a complete description of all necessary cohomology classes. Up to now we have obtained only partial results expressing the desired cohomology classes in the Hurwitz spaces in terms of so-called ``basic'' classes related to the singularities and relative Chern classes. However, already these partial results led to previously unknown enumeration formulas.
We hope that our approach will find a much wider domain of application and consider the Hurwitz problem only as an important example where necessary tools useful in general situation can be developed.

[1] T. Ekedahl, S. K. Lando, M. Shapiro, A. Vainshtein {\it Hurwitz numbers and intersections on moduli spaces of curves}, Invent. Math. {\bf 146} (2001), no. 2, 297--327.
[2] M. E. Kazarian {\it Multisingularities, cobordisms, and enumerative geometry}, Russ. Math. Surv. {\bf 58} (2003), no. 4, 665--724.
[3] M. E. Kazarian, S. K. Lando, {\it On intersection theory on Hurwitz spaces}, Izv. Ross. Akad. Nauk Ser. Mat. {\bf 68} (2004), no. 5, 91-122
[4] S. K. Lando, D. Zvonkine, {\it Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces)}, preprint MPI 2003 - 48, Bonn