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This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin.
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English
English [CC]
FREE
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Description
Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.
Course content
- Smooth manifolds, geometry of foliations, and symplectic structure. Unlimited
- Comments on previous lecture, symplectic manifolds, and Poisson geometry Unlimited
- Almost complex structure, Hermitian structure, integrability of J, forms on a complex manifold, and Dolbeault cohomology. Unlimited
- Geometry of V+V*, linear Dirac structures, and generalized matrices. Unlimited
- Spinors, the spin group, a bilinear pairing on spinors, and pure spinors Unlimited
- Generalized Hodge star, and spinors for TM+T*M and the Courant algebroid. Unlimited
- Exact Courant algebroids, and Severa’s classification of exact Courant algebroids. Unlimited
- Dirac structures, and geometry of Lie groups. Unlimited
- Bilinear forms on groups. Unlimited
- Integrability, Dirac maps, and manifolds with Courant structure. Unlimited
- Integrability and spinors, and Lie bialgebroids and deformations. Unlimited
- Generalized complex structures and topological obstructions, intermediate cases, spinorial description, and introduction to Hermitian geometry. Unlimited
- Generalized Kahler geometry. Unlimited
- Generalized Kahler geometry, and Hodge theory on generalized Kahler manifolds. Unlimited
- Generalized complex branes of rank 1. Unlimited
- Linear algebra, and T-duality. Unlimited
Instructor
Massachusetts Institute of Technology
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