International conference
Combinatorial Methods in Physics and Knot Theory

The cohomology ring of the (compactified) moduli spaces of algebraic curves has extremely complicated structure and there is no hope to get its complete description. It happens, however, that cohomology classes which are interesting from the point of view of applications form a relatively small subring of the whole cohomology ring and its description is much simpler. A version of this subring called the tautological ring have been introduced by Faber and Pandharipande. We suggest another version of the tautological ring. Our ring is even smaller than that of Faber and Pandharipande. It has much nicer functorial properties: most of relations used to compute intersection numbers in this ring are independent of the particular moduli space (the genus of curves or the the number of marked points). This leads to a very efficient algorithm for the computation of particular intersection numbers. As another consequence, we obtain a number of closed formulae for some important families of Hodge integrals.