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Last updated:

November 7, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

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.

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 Curriculum

  • 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

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Massachusetts Institute of Technology