The objective is to study the formulation of geometry in a Courant algebroid setting. The central object that will be carefully constructed is the generalized complex structure; the student will present a coherent yet comprehensive resume of the topic. Finally, it will be shown how such notion serves as a generalization of various geometries with physical relevance and how the discussed structures behave upon reduction onto constrained systems.
Aspects of generalized geometry have emerged in both mathematical and physical contexts during the last two decades. A cohesive theory of generalized complex geometry has been thoroughly formulated by Nigel Hitchin and developed by Marco Gualtieri and others. In particular, Hitchin and Gualtieri have shown how symplectic and complex geometry can be viewed as special complementary cases of a more general unifying structure, the complex generalized geometry. Today, intriguing interpretations and applications of these generalized structures arise, in particular, in field theory and strings.
