Math 319, Topics in Algebra: Lie Groups, fall 2007

Harriet Pollatsek, 400 Clapp, X 2341, hpollats@mtholyoke.edu

MW 8:35 - 9:50, F 9:00 - 9:50, 420 Clapp

The study of Lie groups (named for the Norwegian mathematician Sophus Lie) provides rich connections to many parts of mathematics and important applications to physics and chemistry. Topics include symmetries of real vector spaces, linearization, one-parameter subgroups and the exponential map, Lie algebras and adjoint maps, and classical matrix groups over other fields.

Prerequisites: linear algebra and calculus.

COURSE OUTLINE

1. Classical groups. Groups of symmetries of real vector spaces (Euclidean and Lorentz). Briefly: complex numbers, the unitary groups, and the quaternions. Problem Sets 1 - 5.

2. Linearization. Tangent spaces and differentials. Problem Sets 6 - 7.

3. One-parameter subgroups and the exponential map. (And another way of thinking about differential equations.) Problem Sets 8 - 11.

4. Lie algebras and adjoint maps. Problem Set 12.

5. Classical matrix groups over other fields. The complex numbers and

finite fields; also, quaternions and octonians. Problem Sets 13 - 14.

There is no text, but notes and photocopied readings will be provided. There is a $10 charge for photocopies, billed to your MHC account (or payable directly to Ms. Kamins in 415B if you are not an MHC student). If there's money left, we'll use it toward an end of term party!

SOFTWARE We will make use of Maple for calculations in class and on assignments. Maple is available on the computers in 401, 420 and 422 Clapp and in some other MHC computer labs. You don’t need to purchase it, but if you choose to do so, you are eligible for a discount on the student edition, because this course is registered with Maple (code AP29054).

GRADING

Homework assignments (10) 16%

Presentation problems (6) 6%

Three exams 58%

Paper (4-6 pages) 20%

There will be occasional starred challenge problems, which are optional. However, some success with challenge problems will be expected for grades of A or A-.

Working together on homework problems is encouraged, although you should write up your own solutions. There will be 10 homework assignments to be handed in on Fridays, each worth a maximum of 16 points, check plus = 16, check = 12, check minus = 8. Check plus means the homework is good enough to study from; check minus means you should get help and rewrite. If you wish to resubmit a homework, attach the rewritten problems (just fix what needs fixing) to the original and turn it in with the next assignment; I'll count the better score.

PRESENTATIONS:

Any un-starred problem on a problem set that hasn’t been done in class and isn't to be handed in for homework can be used for a presentation. You should present six problems: #1 by September 21, #2 by October 5, #3 by October 26, #4 by November 12, #5 by November 30, and #6 by December 12. Working together on presentation problems is fine, and you can also consult me. “Reserve" presentation problems ahead of time so there are no repeats.

EXAMS

The first two exams will each have a modest in-class, closed book portion (definitions, etc.) worth 50 points and a take-home portion worth 160 points. The third exam will just have a take-home portion worth 160 points. Collaboration is not permitted on exams. Tentative dates:

Exam 1 (through Problem Set 5): take-home handed out 10/3 and due 10/12; in-class 10/12.

Exam 2 (through Problem Set 9): take-home handed out 11/2 and due 11/9; in-class 11/9.

Exam 3 (through Problem Set 14): take-home handed out 12/7 and due at the end of the exam period, noon 12/19.

It is possible to rewrite exams 1 and 2, and the average of the two scores will be recorded.

PAPER

There is a separate, detailed handout on the paper assignment, including a list of possible topics and suggested references. You should choose your topic by 10/24 (10/22 will be a workday on choosing a topic -- no class), submit an outline by 11/12, and submit the paper by 6:00 pm

Tuesday 12/4.