Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n -point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of n -point functions in terms of summation over binary trees.

In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, ie the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete Khler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level m structure is also injective.

In this note we derive some consequences from the Mario-Vafa formula and the cut-and-join equation, these include unified simple proofs of the \lambda_g conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mario-Vafa formula.

In this paper, we study effective recursion formulae for computing intersection numbers of mixed \psi and \psi classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani's recursion and prove a recursion formula of higher WeilPetersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves.

We obtain a system of relations between linear Hodge integrals. As an application, we show that its first non-trivial relation implies the Witten's Conjecture/Kontsevich Theorem.