Arnold's seminar, Moscow University
February 18, 2003

S.K.Lando
"On conjectures forming the base of the geometry of
spaces of rational functions"

Recent results belonging to the speaker and D.Zvonkine, concerning
the Hurwitz problem about the enumeration of ramified coverings
of the 2-sphere with prescribed ramification types, will be described.
(In the modern setting, the problem is equivalent to the computation
of the quantum cohomology of the projective line.) This problem
is reduced, in a now standard way, to a problem about intersection
indices in spaces of rational functions. The latter problem can be
solved provided that we know the cohomology rings of these spaces,
as well as the elements of the rings represented by discriminant strata.
Known answers, as well as conjectures simplifying computations
and presumable final answers for the case where the covering
surface also is the sphere will be presented.