M. Hamilton and D. Kotschick: Symplectic and bi-Lagrangian structures

• Time and place: Thu and Fri 10-12 in HS B 251
• Lecture notes
• Contents: This course is about the geometry and topology of manifolds equipped with a symplectic form and a pair of complementary Lagrangian foliations. Such structures appear naturally in physics in the contexts of classical mechanics and of geometric quantisation. On the mathematical side they lie at the crossroads of symplectic geometry and of the theory of foliations, and therefore have a very rich and intricate geometry. There are several names used for these structures in the literature, such as bi-Lagrangian structures, and para-Kähler or Künneth structures.
  We will develop some of the background on symplectic structures and on foliations, as well as some of the relevant differential-geometric tools, such as symplectic connections and pseudo-Riemannian metrics of neutral signature. We will then discuss the geometry of bi-Lagrangian manifolds, as well as the global topology of compact examples.
• Intended audience: All students of mathematics and/or physics at the master or doctoral level who are interested in geometry and have a good knowledge of the basic facts about smooth manifolds. Some previous exposure to either symplectic geometry or to the theory of foliations is useful, but not required.
• References
  M. Audin, A. Cannas da Silva, E. Lerman, Symplectic geometry of integrable Hamiltonian systems, Lectures delivered at the Euro Summer School held in Barcelona, July 10-15, 2001. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2003.
  A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001.
  A. Weinstein, Lectures on symplectic manifolds, CBMS Lecture Notes, Vol. 29, Amer. Math. Soc., Providence, R.I. 1977.
  
• Examination: There will be an oral exam in July. Details about the registration for the exam will be given during the lectures. In the mathematics master programme, students can get credit for modules 42.2 or 46.2. In the TMP master programme, this course counts as a special module relevant to areas B, C and D.