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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Suppose A and B are normed division algebras, i.e. R,C,H or O, we introduce and study Grassmannians of linear subspaces in (A ⊗ B)n which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on (A ⊗ B)n. We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.

We compute certain open Gromov–Witten invariants for toric Calabi–Yau threefolds. The proof relies on a relation for ordinary Gromov–Witten invariants for threefolds under certain birational transformation, and a recent result of Kwokwai Chan.

We study the geometry of the Grassmannians of symplectic subspaces in a symplectic vector space. We construct symplectic twistor spaces by the symplectic quotient construction and use them to describe the symplectic geometry of the symplectic Grassmannians.