MCMASTERUNIVERSITY

Math 766 : Lie Groups

Winter Term, 2010-2011

Course Outline

Instructor: Dr. M. Wang

Office: Hamilton Hall 321

Phone: 23405 ; e-mail:

Office hours:Wednesdays 1:30 pm – 3:30 pm, or by appointment

Course Objectives:

This is an introduction to Lie group theory through applications and examples. We will

discuss the correspondence between Lie groups and Lie algebras, general structure theory,

elements of representation theory, and homogeneous spaces. Applications to geometry,

topology, and physics will be specially emphasized.

Prerequisites:

A firm command of linear algebra and multivariable calculus; familiarity with basic topological

notions such as metric spaces, connectedness, compactness; willingness to learn something about

smooth manifolds as the course progresses

References:

There is no formal textbook for the course, but as general references, I recommendone of the following:

[FH]W. FultonJ. Harris: Representation Theory, A First Course, Graduate Texts in Mathematics, Vol.129, Springer-Verlag (1991 or later).

[BT] T. Brocker & T. tom Dieck: Representations of Compact Lie Groups, Graduate Texts

in Mathematics, Vol. 98 , Springer-Verlag (1985).

[DK] J. Duistermaat & J. A. C. Kolk: Lie Groups, Universitext, Springer-Verlag, (1999).

[V] V. S. Varadarajan: Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in

Mathematics, Vol. 102, Springer-Verlag (1984).

The following is a further list of references which are devoted tomore specialised topics:

[H] S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies

in Math., Vol. 34, American Math. Soc., (2001).

[S] S. Salamon: Riemannian Geometry and Holonomy Groups, Pitman Research Notes in

Mathematics, Vol 201, Longman Scientific & Technical, (1989).

[W] J. Wolf: Spaces of Constant Curvature, Sixth edition, AMS-Chelsea, (2011).

[O] P. Olver: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics,

Vol. 107, Springer-Verlag, (1993).

Course Work:

1. Written assignments—There will be 3 assignments which will be handed out in class.

2. Class presentation—Each registered student must sign up for one class presentation, which should take about 45-60 minutes.The presentations will be scheduled by private discussion and will occur throughout the term. A list of possible topics will be announced in class together with suitable references. Students then prepare their talk and arrange to meet with mebefore their presentation. Class presentations will be evaluated according to the correctness of the mathematical content, the clarity of the talk, and the quality of the illustrative examples used.

3. Take-home final exam

Grading System

Your course mark will be composed of

  • 45% homework assignments
  • 15% class presentation
  • 40% take-home final exam

Official Policies

Academic dishonesty consists of misrepresentation by deception or by other fraudulent means, and can result in serious consequences, e.g., the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: “Grade of F assigned for academic dishonesty”), and/or suspension or expulsion from the university.

It is your responsibility to understand what constitutes academic dishonesty. For information on the various kinds of a academic dishonesty please refer to the Academic Integrity Policy with URL

The following illustrates only three forms of academic dishonesty:

1.Plagiarism, e.g., the submission of work that is not one’s own or for which other credit has been obtained. Discussions about homework are allowed and are generally beneficial. However, you must write up solutions of the assignment problems by yourself and in your own words. Copying with minor changes (e.g., with symbols changed or with slightly different wording) from solutions prepared by another person or publication, in whatever format, will be dealt with as an act of plagiarism.

2. Improper collaboration in group work.

3. Copying or using unauthorized aids during tests and examinations.

Regarding collaboration in written assignments: Discussion about assignments is allowed and is generally beneficial. However, you must write up solutions of the assignment problems by yourself and in your own words. Copying with minor changes (e.g., with different symbols or slightly different wording) from solutions prepared by another person, publication or website, in whatever format, will be dealt with as an act of plagiarism.

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