Arnold's seminar
April 26, 2005

M. Kazarian
An algebro-geometric proof of Witten's conjecture.

I mean `small' conjecture of 1991 proved by Kontsevich in 1993.
It predicts some intersections numbers on the moduli space M_{g,n}
of complex curves with marked points. Though the formulation of the
statements is algebro-geometric, all known proofs (there are at least
tree published ones) use in an essential way real topology.

I present a rather simple purely algebro-geometric way to the
computation of these numbers obtained during our discussions
with S. Lando, D. Zvonkin, and S. Shadrin. The approach is based on the
classical now so called ELSV (Ekeahl-Lando-Shapiro-Vainshtein)
formula (1998) that expresses certain Hurwitz numbers (the numbers
of ramified coverings of a sphere with prescribed ramifications)
as certain Hodge integrals over the moduli space. Strangely enough
a simple fact that this formula can be `inverted' to express intersection
numbers via Hurwitz ones have not been noticed for a long time.

As a result, the proof of the conjecture is reduced to the verification
of the coincidence of two explicit formulae for the generating
function of these numbers. Using computer, I checked this for
several dozen of particular numbers, however, at the moment I have
no formal proof.

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