We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of n-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of Gromov-Witten invariants.

This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature. The same conclusion is obtained for n-dimensional hyperbolic space as well.

In this paper, we show that any compact K\"ahler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K\"ahler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:\C\> X$.

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We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.