IGA Lecture Series

IGA Lecture Series 2011

There will be several distinguished IGA Lecturers spread throughout the year, who will give lecture series on current topics of intense international interest in Geometry and its Applications.

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For the IGA Lecturers in 2010, click here.

Eckhard Meinrenken (University of Toronto, Canada)

Group-valued moment maps and moduli spaces of flat G-bundles

He will be speaking at the IGA/AMSI Workshop, Group-valued moment maps with applications to mathematics and physics

5 lectures (2 hours each), September 5-9, 2011. The lectures will be held at the Conference Room 7.15, Level 7, Innova 21 building, University of Adelaide.

Lecture 1: Introduction to G-valued moment maps

The theory of quasi-Hamiltonian G-spaces with G-valued moment maps has its origins in 2-dimensional gauge theory. Its features are similar to the usual Hamiltonian theory, but with interesting modifications. We will give an overview of the theory, with discussions of a convexity theorem and a Kirwan surjecticity theorem.

Lecture 2: Dirac Geometry and Witten's volume formulas

Dirac geometry treats 2-forms and bivector fields within a common framework. We will explain how this leads to a conceptual approach to G-valued moment maps. As an application, we will construct Liouville volume forms, leading to a proof of Witten's volume formulas for moduli spaces of flat G-bundles over surfaces.

Lecture 3: Dixmier-Douady theory and pre-quantization

Dixmier-Douady bundles provide geometric realizations of integral degree three cohomology classes over a space. We will use these bundles to construct distinguished "twisted Spin-c structures" on quasi-Hamiltonian G-spaces, and also define pre-quantizations in similar terms.

Lecture 4: Quantization of group-valued moment maps

We will review Rosenberg's definition of twisted K-homology in terms of Dixmier-Douady bundles and the Freed-Hopkins-Teleman theorem on the twisted K-homology of Lie groups. We then define the quantization of group-valued moment maps as push-forwards in twisted K-homology.

Lecture 5: Application to Verlinde formulas

The quantization of a quasi-Hamiltonian space is computable via localization. In conjunction with a `quantization commutes with reduction' theorem, this leads to the symplectic version of the Verlinde formulas for moduli spaces of flat G-bundles.

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