Lie Groups

M390C. David Ben-Zvi. Fall Semester 2009. Tue-Th 9:30-10:45 RLM 10.176.

Office Hours:

W 2-3,4-5 and by appointment

Notes:

My notes for the class can be found here: Part 1, Part 2, Part 3, Part 4(to be updated as we go along).

Bibliography:

R. Carter, G. Segal and I.MacDonald: Lectures on Lie Groups and Lie Algebras. London Math. Society Student Texts 32. The lectures by Segal are a beautiful overview of the fundamental ideas of Lie groups and algebras, with geometry and examples emphasized.
W. Rossmann: Lie Groups: an introduction through linear groups (very concrete introduction to Lie groups and representations)
J.F. Adams: Lectures on Lie groups (concise and very clearly written)
F. Warner, Foundations of Differentiable Manifolds and Lie Groups (thorough introduction to differential topology basics, including general Lie theory)
D. Bump, Lie groups (very fast and far-reaching course on Lie theory)

I will not follow any text precisely, but you will find it useful to have a text to supplement the lectures. I will recommend several other sources during the course. We will discuss many examples of Lie groups and Lie algebras, the relations between the two, key topological and geometric features of Lie groups and homogeneous spaces, and the elements of the representation theory of compact Lie groups.

Topics covered (tentative, suggestions welcome):

Basic examples and definitions of Lie theory
Lie groups and covering spaces
Lie algebras
Relating Lie groups and Lie algebras; the exponential map
Homogeneous spaces
Solvable vs. Semisimple Groups
Structure of compact Lie groups (maximal tori, Weyl groups etc.)
Integration on compact Lie groups
Fourier theory and the Peter-Weyl theorem
Geometric representation theory: the Borel-Weil theorem
Principal bundles and classifying spaces (time permitting)

Homework:

I will hand out problem sets, and you can and should find plenty of problems of your own from the lectures. I urge you to immediately form study groups and to discuss the problems and lectures together, as well as individually. The homeworks will be graded with checks only, and will count for %20 of the final grade.

Homework assignments

Projects and Grades:

Those of you who are registered for the course will work in groups of one or two on a project. At the end of the semester each group will hand in a (joint) paper. I am very open about the topics and will suggest some as we go along. Before starting on a project, please come talk to me about the topic. I hope you will get started by late October at the latest. Grades are mostly (%80) based on the projects. Grades will be assigned with plusses and minuses.

Some suggestions for projects (more to come):

The exceptional groups and the octonions
Bott periodicity and Morse theory on classical Lie groups
Differential geometry of symmetric spaces
The Heisenberg group in Fourier theory and harmonic analysis
Twistors and geometry of Minkowski space
Invariant theory in algebra & geometry via the classical groups
Mostow rigidity in hyperbolic geometry
Margulis superrigidity
Cohomology of classifying spaces for Lie groups and characteristic classes
The Yang-Mills equations and connections on principal bundles
Berger's classification of holonomy groups in differential geometry
The orbit method for describing representations of nilpotent groups
Representation theory in quantum mechanics
Loop groups
The Borel-Weil-Bott theorem: representations and cohomology of flag manifolds
Representations of SL2R
Quantum groups
Finite groups of Lie type
p-adic groups
Formal groups
Lie groups and systems of differential equations
Modular and automorphic forms
Equivariant cohomology
Ergodic theory on Lie groups
Vector fields on spheres in homotopy theory

Office Hours:

I will hold regular office hours, with a minimum of an hour every week, and more if there is demand. I also encourage you to come talk to me about various questions related to the class, and to email me for appointments outside of the official office hours.

Prerequisites:

I will assume familiarity with basic notions of topology, such as manifolds, covering spaces, fundamental groups, and tangent bundles, as they are covered in the prelim topology sequence, and with basic group theory. We will however work mostly in concrete settings rather than developing the theory in its most abstract context, so an in depth technical familiarty with these notions is not required.

Seminars:

I encourage you to at least sample the weekly geometry seminars. The main Geometry Seminar is Thursdays at 3:30. Speakers are encouraged to be expository during the first hour, and this usually makes that seminar more accessible. There is a regular Geometry and String Theory seminar on Wednesdays at 12:00, and an occasional GADGET lunch seminar Tuesdays at 12:30. All seminars are in RLM 9.166. You shouldn't expect to understand everything at a research seminar, or even in some cases to understand very much. But only by attending seminars will you learn about a field: its problems, techniques, style, priorities, personality and personalities, etc. I cannot urge you strongly enough to sample all of our many departmental seminars.