We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the <i>n</i>-point functions for Witten9s classes, an effective recursion formula to compute higher WeilPetersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed and intersections.

Soit G un groupe algbrique complexe quasi-simple et G/P une varit de drapeaux partiels. La projection sur G/P des varits de Richardson (de la varit des drapeaux complets) forment une stratification de G/P. Nous montrons que les relations dadhrence des varits de Richardson projetes correspondent celles dun certain sous-ensemble de varits de Schubert sur la varit de drapeaux affine de G. Nous comparons aussi les classes de cohomologie quivariante et de K-thorie de ces deux stratifications. Notre travail gnralise celui de Knutson, Lam et Speyer pour la grassmannienne de type A.

This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.

Let R be a non-discrete rank one valuation ring of characteristic p and let E be any discrete valuation ring, we prove the ring of E-Witt vectors over R has uncountable Krull dimension without assuming the axiom of existence of prime ideals for general commutative unitary rings.

We give a short proof of Penner-Grushevsky-Schumacher-Trapanis large genus asymptotics of Weil-Petersson volumes of moduli spaces of curves. We also study asymptotic expansions for certain integrals of pure \psi classes and answer a question of Mirzakhani on the asymptotic behavior of one-point volume polynomials of moduli spaces of curves.