## Welcome

This course is a solid introduction to Generalized Geometry, a new viewpoint to geometric structures with interest both in mathematics and theoretical physics. In the first part of the course we will look at generalized linear algebra (the linear algebra of the vector space V+V*, and classical geometry. This will give us the confidence to work, in the second part of the course, on the basics of generalized geometry, not as mere spectators, but as active actors. The course takes place on Sundays 13:15-16:00 at If you want to know more, please, check the sections Syllabus, Assignments and Lecture Notes, or contact me by email. The official course information can be found at the Feinberg Graduate School. |

## Summary of the lectures

05 Jul 2018. We looked at the structure groups involved in generalized geometry, showed an example of a 4-manifold that is neither symplectic nor complex, but still generalized complex, introduced the definition of generalized Kahler manifolds and mentioned their relation with mirror symmetry.

01 Jul 2018. We looked at the type of generalized complex structures, saw examples of structures of different type and show the phenomenon of type-change with examples both in Dirac and generalized complex geometry.

28 Jun 2018. We looked at the orthogonal symmetries of the Courant bracket, the generalized diffeomorphisms, and we described the involutivity of a maximally isotropic subbundle in terms of differential forms.

24 Jun 2018. We saw that Poisson structures correspond to symplectic foliations and that Dirac structures correspond to presymplectic foliations.

17 Jun 2018. We reviewed the definitions of the exterior derivative, the Lie derivative and the Lie bracket. We recalled the integrability of complex and symplectic structures, and used it to define the Dorfman bracket on T+T*. We proved its properties and defined Courant algebroids. We also started to discuss Poisson structures.

10 Jun 2018. We described the forms giving linear generalized complex structures, defined the type and describe what any linear generalized complex structure looks like. We started with Part 3 by recalling some basic geometric notions, as the integrability of an almost complex structure.

3 Jun 2018. We looked at which forms give, by their annihilators, Dirac structures. We reviewed the Clifford algebra, the Clifford group and showed that the Pin group is a double cover of the orthogonal group.

27 May 2018. We talked about linear generalized complex structures, the Clifford algebra and forms as spinors.

24 May 2018. Basic geometry session.

13 May 2018. We gave a complete description of maximally isotropic subspaces of V+V*, we introduced the group of generalized linear transformations, and talked about annihilators of forms.

22 Apr 2018. We continued with generalized linear algebra, looking at maximally isotropic subspaces of V+V*, examples, etc.

15 Apr 2018. We reviewed the tensor, exterior and symmetric algebras of a vector space, looked at our structures from this viewpoint and started with generalized linear algebra.

8 Apr 2018. We finished with linear symplectic structures and talked about linear complex structures.

25 Mar 2018. We presented the course and talked about linear symplectic structures