Moscow-Petersburg seminar on Low-dimensional mathematics April 23, 2004 S.K.Lando (Moscow) Applications of global theory of singularities to intersection theory of Hurwitz spaces and universal polynomials We suggest a new approach to the intersection theory on Hurwitz spaces, that is spaces of meromorphic functions on algebraic curves. This approach is based on the theory of universal polynomials originating in the work by R.~Thom in early 60ies and recently developed by M.~Kazaryan. This theory allows one to express the cohomology classes Poincar\'e dual to loci of singularities of a general holomorphic mapping $f:M\to N$, of given type, in terms of universal polynomials in the relative Chern classes of~$f$. As a result, we shed a fresh light on the structure of cohomology of Hurwitz spaces and obtain new enumeration results in the framework of the Hurwitz problem concerning enumeration of ramified coverings of the 2-sphere, with prescribed ramification type. We hope that our approach will find a much wider domain of application and consider Hurwitz spaces as an important example where necessary tools useful in general situation can be developed. --- http://www.pdmi.ras.ru/~lowdimma