Mathematics

Topics to be covered:

1. Symplectic and Kähler structures;
2. Hermitian structures;
3. Almost-complex structures and integrability;
4. Obstructions to the existence of an almost-complex structure;
5. Cohomology of almost-complex manifolds;
6. Courant bracket and Dirac structures;
7. Generalized complex structures;
8. Submanifolds of generalized complex structures;
9. Generalized Kähler structures.

Bibliography:

• R. Bryant, On the geometry of almost-complex manifolds. Asian J. Math. 10 (2006), 561 - 605;
• M. Gualtieri, Generalized complex geometry. arXiv:math/0401221, January 2004;
• M. Gualtieri, Generalized complex geometry. Ann. of Math. 2 (2011), 75--123;
• N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281 -308;
• N. Hitchin, Lectures on generalized geometry. arXiv:math.DG:1008.0973, August 2010.