# Symplectic Geometry Seminar

Wednesday 3:30pm-4:30pm VV B139

date speaker title host(s)
Sept. 21st Ruifang Song The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties
Sept. 28st Ruifang Song The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties(continued)
Oct. 5th Dongning Wang Seidel Representation for Symplectic Orbifolds

## Abstracts

Ruifang Song The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties

Abstract

We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which have played important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties. In general, if X is a Fano variety, L is the anticanonical line bundle and G=Aut(X), this construction can be used to study the Picard-Fuchs system of Calabi-Yau hypersurfaces/complete intersections in X.

Dongning Wang Seidel Representation for Symplectic Orbifolds

Abstract

For a symplectic manifold, Seidel representation is a group morphism from $\pi_1(Ham(M,\omega))$ to the multiplication group of the quantum cohomology ring $QH^*(M,\omega)$. With this morphism, once given enough information about $\pi_1(Ham(M,\omega))$, one can get compute $QH^*(M,\omega)$. On the other side, as suggested by physicists, there is mirror symmetry between orbifolds and its mirror. To check this, one would like to compute the quantum cohomology of symplectic orbifolds which is defined by Chen and Ruan. We generalize Seidel representation in order to compute quantum cohomology for symplectic orbifolds. Thus we need to generalize all the terminologies used to construct Seidel representation. For a symplectic orbifold $(\mathcal{X},\omega)\lt math\gt , we define \lt math\gt \pi_1(Ham(\mathcal{X},\omega))$, Hamiltonian orbifiber bundles, and sectional pseudoholomorphic orbicurves into the bundle. In this talk, I will explain these definitions and use them to construct Seidel representation.
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