Math 597C, Fall 2010


Course Title: Compact Groups and Symplectic Manifolds

Instructor: Nigel Higson

Class Meeting Times: Mondays, Wednesdays and Fridays, 2:30-3:20, in 112 Sackett

Office Hours:  Mondays and Tuesdays, 3:30-4:30, in 228 McAllister

The course will consist of two parts.  In the first, some foundational material on manifolds, Lie groups and symplectic geometry will be presented. In the second, several more advanced topics will be examined.  At the very beginning the pace will be brisk, I hope, since I expect that participants in the course will have encountered smooth manifolds before (to the extent of, say, several chapters in Spivak's Comprehensive Introduction to Differential Geometry, Vol. 1).  The pace will pick up again in the second part of the course, but in between we shall proceed more slowly and thoroughly.

To get a feeling for where the course will be heading, you might browse through this paper:

M.F. Atiyah, Angular momentum, convex polyhedra and algebraic geometry.

Proc. Edinburgh Math. Soc. (2) (1983) 26 (2) pp. 121-133.

A complete reading list will be supplied soon, along with further details about the course.

Part I

Smooth Manifolds and Differential Calculus

Tangent bundle, vector fields, flows

Lie brackets, Frobenius theorem

de Rham algebra

Lie derivatives, contractions, Cartan homotopy formula

Lie Groups

Matrix Lie groups and abstract Lie groups

Invariant vector fields

Lie bracket

Lie algebras

Exponential map

Lie group actions

Symplectic Manifolds

Symplectic linear algebra

Symplectic manifolds

Poisson bracket, Hamiltonian vector fields

Moser trick, Darboux theorem

Cotangent bundle, co-adjoint orbits

Symplectic manifolds in classical mechanics

Hamiltonian Group Actions

Moment maps

Symplectic reduction

Aspects of geometric quantization

Part II

Convexity Theorem

Schur-Horn theorem

Morse theory

Local convexity

Global convexity

Duistermaat-Heckman Formula

Equivariant cohomology

Stationary phase

Fixed point formula

Duistermaat-Heckman measure

Nonabelian Localization

Witten's nonabelian localization formula

Equivariant cohomology with generalized coefficients

Proof(s) of the localization formula

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Nigel Higson - 2010