We give a natural filtration F on QH(G/B), which respects the quantum product structure. Its associated graded algebra GrF(QH(G/B)) is isomorphic to the tensor product of QH(G/P) and a corresponding graded algebra of QH(P/B) after localization. When the quantum parameter goes to zero, this specializes to the filtration on H(G/B) from the Leray spectral sequence associated to the fibration P/B ! G/B −! G/P.

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By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.

In this article we study compact K¨ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition.

For almost all Riemannian metrics (in the C1 Baire sense) on a closed manifold M^{n+1}, 3 \leq n + 1 \leq 7, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of a result by Irie and the first two authors, that established density of minimal hypersurfaces for generic metrics. The main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors .