We consider a measurable matrix-valued cocycle A: Z+ X Rd d, driven by a measurepreserving transformation T of a probability space (X,, ), with the integrability condition log+ A (1,) L1 (). We show that for -ae x X, if limn 1 n log A (n, x) v= 0 for all v Rd\{0}, then the trajectory {A (n, x) v} n= 0 is far away from 0 (ie lim supn A (n, x) v> 0) and there is some nonzero v such that lim supn A (n, x) v v. This improves the classical multiplicative ergodic theorem of Oseledec. We here present an application to linear random processes to illustrate the importance.

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.

We define the quantum correction of the Teichmller space \mathcal {T} of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichmller space \mathcal {T} is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image \mathcal {T} of the Teichmller space \mathcal {T} under the period map \mathcal {T} is an open submanifold of a globally Hermitian symmetric space \mathcal {T} of the same dimension as \mathcal {T} . Finally, for Hyperkhler manifold of dimension \mathcal {T} , we find both locally and globally defined families of \mathcal {T} and \mathcal {T} -classes over the Teichmller space of polarized Hyperkhler manifolds.

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on (4 r 1) dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on (4 r 1) dimensional spin manifolds and some congruent formulas on characteristic number for (4 r 1) dimensional spin c manifolds.

We establish the asymptotic expansion of certain integrals of classes on moduli spaces of curves , when either the or goes to infinity. Our main tools are cut-join type recursion formulae from the WittenKontsevich theorem, as well as asymptotics of solutions to the first Painlev equation. We also raise a conjecture on large genus asymptotics for -point functions of classes and partially verify the positivity of coefficients in generalized Mirzakhanis formula of higher WeilPetersson volumes.