Generalized geometry is a novel approach to geometric structures pioneered by Hitchin in 2003. Developed further by Gualtieri, Cavalcanti and others, it soon became an active topic catching the interest and bringing together the expertise of geometers and theoretical physicists. For example, generalized complex geometry provides a uniform setting for complex and symplectic structures, which made it suitable for the study of mirror symmetry and as a geometric explanation of the extended complex deformation space.
In this introductory course we will present the basics of this topic starting from basic linear algebra and focusing on the study of Dirac structures, which were introduced in 1990 and have applications in Mechanics, and generalized complex structures, the best and most useful instance of generalized geometry. After introducing linear complex/symplectic/Poisson structures, we will define some generalized concepts in the context of linear algebra. We will then move to classical geometry, focusing on the meaning of integrability, and finally deal with generalized geometry.
We will cover the following topics:
I. Linear algebra
- Linear complex structures.
- Linear symplectic structures.
- Linear Poisson structures.
II. Generalized linear algebra
- The vector space V+V*: the canonical pairing.
- Isotropic subspaces and orthogonal transformations.
- Linear Dirac structures.
- Differential forms as spinors and the Clifford algebra.
- Linear generalized complex structures and the real index.
- Almost complex structures and non-degenerate 2-forms.
- Integrability: complex and symplectic structures.
- Poisson structures.
IV. Generalized geometry
- The Dorfman bracket and the Courant algebroid TM+T*M.
- The group of generalized diffeomorphisms.
- Dirac structures: geometric interpretation as foliations.
- Type-change structures.
- Integrability in terms of spinors.
- Generalized complex structures.
- Type change in generalized complex geometry.
- A generalized complex manifold that admits neither complex nor symplectic structures.
- Other topics, depending on the interest and time: twisted versions, other generalized tangent bundles, generalized riemannian geometry, deformation of complex and generalized complex structures, applications to Physics, possible research directions.