Geometric Quantization
This lecture course (2 hours a week, Master, V5D3) will explain geometric efforts to "quantize a classical system". Nevertheless, we will only briefly talk about the physical background, but will use this theory as a tool to understand more about Spinc manifolds with a Lie group action. For this we will define the Spinc quantization which is the equivariant index of a Dirac operator
We will start by explaining the background material (Spinc structures, Dirac operators, equivariant cohomlogy) and then prove and discuss to remarkable theorems:
- Additivity under cutting: We cut an manifold along a hypersurface into two manifolds (where we "destroy" the boundary by a special construction), then the resulting quantizations add up to the original one.
- Quantization commutes with reduction: With the help of a hypersurface, we can construct a reduced manifold of dimension 2m-2. Und sensible assumptions on the hypersurface, this reduction procedure commutes with the quantization.
Prerequisites: "Globale Analysis I", basic knowledge of vector bundles, principal bundles and connections
Useful techniques (knowledge not assumed): Spinc structures, Dirac operators, Representations of compact Lie groups
Furthermore, you will learn something about applications of the Atiyah-Singer-index theorem and the Lefschetz fixed point formula (we do not assume prior knowledge about these topics either; instead we will discuss ideas and applications while giving proper definitions but nearly no proofs).
For more information, please contact me (fmeierXmath.uni-bonn.de).
Lecture notes for are provided under the following link (last update: 20th December 2011):
Lecture notes
The course will take place every Friday from 10 a.m. to 12 p.m. in 1.007 (Landwirtschaftskammer). If this is uncomfortable for you, please contact me as well.
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