# Symplectic Geometry Seminar: Difference between revisions

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Abstract | Abstract | ||

For a symplectic manifold <math>(M,\omega)</math>, Seidel representation is a group morphism from <math>\pi_1(Ham(M,\omega))</math> to the multiplication group of the quantum cohomology ring <math>QH^*(M,\omega)</math>. With this morphism, once given enough information about <math>\pi_1(Ham(M,\omega))</math>, one can get compute <math>QH^*(M,\omega)</math>. 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 <math>(\mathcal{X},\omega)</math>, we define <math>\pi_1(Ham(\mathcal{X},\omega))</math>, 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 for orbifolds. | For a symplectic manifold <math>(M,\omega)</math>, Seidel representation is a group morphism from <math>\pi_1(Ham(M,\omega))</math> to the multiplication group of the quantum cohomology ring <math>QH^*(M,\omega)</math>. With this morphism, once given enough information about <math>\pi_1(Ham(M,\omega))</math>, one can get compute <math>QH^*(M,\omega)</math>. 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 <math>(\mathcal{X},\omega)</math>, we define loop of Hamiltonian diffeomorphisms, <math>\pi_1(Ham(\mathcal{X},\omega))</math>, 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 for orbifolds. | ||

==Past Semesters == | ==Past Semesters == | ||

*[[ Spring 2011 Symplectic Geometry Seminar]] | *[[ Spring 2011 Symplectic Geometry Seminar]] |

## Revision as of 08:38, 30 September 2011

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

- If you would like to talk in the seminar but have difficulty with adding information here, please contact Dongning Wang

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 [math]\displaystyle{ (M,\omega) }[/math], Seidel representation is a group morphism from [math]\displaystyle{ \pi_1(Ham(M,\omega)) }[/math] to the multiplication group of the quantum cohomology ring [math]\displaystyle{ QH^*(M,\omega) }[/math]. With this morphism, once given enough information about [math]\displaystyle{ \pi_1(Ham(M,\omega)) }[/math], one can get compute [math]\displaystyle{ QH^*(M,\omega) }[/math]. 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 [math]\displaystyle{ (\mathcal{X},\omega) }[/math], we define loop of Hamiltonian diffeomorphisms, [math]\displaystyle{ \pi_1(Ham(\mathcal{X},\omega)) }[/math], 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 for orbifolds.