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.

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to a sphere.

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

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A Schubert class  in the Grassmannian G(k, n) is rigid if the only proper subvarieties representing that class are Schubert varieties . We prove that a Schubert class  is rigid if and only if it is defined by a partition  satisfying a simple numerical criterion. If  fails this criterion, then there is a corresponding hyperplane class in another Grassmannian G(k0, n0) such that the deformations of the hyperplane in G(k0, n0) yield non-trivial deformations of the Schubert variety . We also prove that, if a partition  contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian G(k0, n0), then there does not exist a smooth subvariety of G(k, n) representing .