We study the CalabiYau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

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Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of A-infinity algebras built of differential forms on the symplectic manifold. We show that these symplectic A-infinity algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these A-infinity algebras are equivalent to the standard de Rham differential graded algebra on certain odd dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic A-infinity algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic A-infinity algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related <i>A</i>-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an <i>analogue of divisor equation</i> in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions <i>W</i>, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of , this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with

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We prove that the distribution of eigenvectors of generalizedWigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.

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No abstract uploaded!

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers.