• Marius Crainic (Univ. Utrecht), The local structure of Poisson manifolds.

Abstract:

The aim is to look at the original paper of A. Weinstein with the same title as the present mini-course, and re-do it using the more recent developments in Poisson Geometry.

• Giovanni Felder (ETH Zurich), Holomorphic differential operators and Riemann-Roch-Hirzebruch formulae.

Abstract :
This course is about new developments in the theory of holomorphic differential operators inspired by the progress in deformation quantization of Poisson manifolds. We will start with the local theory: the algebra of polynomial differential operators, its Hochschild homology and the algebraic Riemann-Roch-Hirzebruch theorem of Feigin-Tsygan. We will then consider holomorphic differential operators acting on sections of holomorphic vector bundles and derive a Riemann-Roch-Hirzebruch type formula for supertraces of the action on cohomology.

• Marco Gualtieri (MIT), An introduction to generalized complex geometry.

Problem sheet for students

Abstract:

I will introduce the main concepts of generalized complex geometry, building many examples along the way. Important issues we will address include the connection to Poisson geometry, the relation to gerbes and Neveu-Schwarz flux, and exotic 4-dimensional examples. We end with a summary of generalized Kahler geometry and the relation to holomorphic Poisson structures.

• Reyer Sjamaar (Cornell Univ.)

Course plan:

1. The moment map in symplectic and algebraic geometry
2. Convexity properties
3. Equivariant cohomology and K-theory
4. Group-valued moment maps

Abstract:

When a Lie group acts on a symplectic manifold and the vector fields that generate the action are Hamiltonian, the Hamiltonian functions can be collected into a map, which is called the moment(um) map of the action.  If the manifold is a complex projective variety and the group actsholomorphically, the properties of the moment map bear a close relationship to the invariant theory of the variety.  This will be the topic of the first lecture.  In the second lecture I will deduce from these facts some of the famous convexity properties of the moment map.  The components of the moment map are Morse-Bott functions, which put strong constraints on the topology of the manifold, in particular its equivariant cohomology and K-theory.  This will be explained in the third talk.  In the fourth talk I will discuss various extensions of these results to other types of geometry, with an emphasis on quasi-Hamiltonian geometry and group-valued moment maps.