Tópicos de Geometria Diferencial: Dirac structures and generalized geometry.

These are the problems sheets of the first part of the course:

Problem Sheet 1: Lie algebroids, their Cartan magic's formula and the Dorfman bracket.

Problem Sheet 2: The Schouten bracket and the way to Courant algebroids.

Problem Sheet 3: More on Dirac structures and the linear algebra of V+V*.

Problem Sheet 4: Generalized Kähler geometry.

• Aulas: Ter/Qui/Sex 10:00-12:00, sala 347.
• Ementa: The course offers an introduction to Dirac structures and generalized geometry: In the first part we will look at how and why the notions of Dirac structure, Courant algebroid and generalized complex structure were introduced. We will focus on explaining the basic concepts and their interplay. In the second part we will move to more specialized topics.
  Part I - Basics:
  • Presymplectic and Poisson structures in Mechanics.
  • Dirac structures.
  • Courant algebroids.
  • Generalized complex geometry.
  Part II: Topics
  • Generalized metrics and generalized Kaehler geometry.
  • Generalized reduction.
  • Normal forms.
  • Supergeometric viewpoint.
• Avaliação: Será baseada em listas de exercicios (parte I) e trabalho (parte II).
• Bibliografia :
  • H. Bursztyn, A brief introduction to Dirac manifolds, Arxiv: 1112.5037.
  • T. Courant, Dirac manifolds, Trans. AMS, 1990.
  • N. Hitchin, Generalized Calabi-Yau manifolds, Quaterly Math. J., 2003.
  • M. Gualtieri, Generalized complex geometry. PhD thesis, Oxford. Published version at Annals of Math., 2011.
  • M. Gualtieri, Generalized Kahler geometry. Comm. Math. Phys., 2014.
  • H. Bursztyn, G. Cavalcanti, M. Gualtieri. Reduction of Courant algebroids and generalized complex structures. Advances in Math, 2007.
  • M. Bailey. Local classification of generalized complex structures. J. Differential Geom., 2013.
  • D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, Contemporary Math 315, 2002.