18.704 Seminar in Algebra, Spring 2008

Representation theory of finite groups

Representation theory is a major mathematical technique for exploiting the presence of symmetry. For example, the structure of the hydrogen atom, one of the fundamental computations in quantum mechanics, is controlled by representation theory. Another way of thinking about representation theory is as a generalization of the theory of eigenvalues and matrices that many people see in a linear algebra course. In recent years, representation theory has become a mathematical bridge connecting many branches of mathematics. In this course, we will study representation theory of finite groups (and compact groups when time permits) and explore its applications to probability, geometry etc.

Time : MWF 11-12pm
Place : 2-146
Instructor : JuLee Kim
Office : 2-281
Phone : 253-7576
e-mail: julee at math.mit.edu
Office hours: MF 12-1pm or by appointment

Texts and references

The main textbook is Linear representations of finite groups by J.-P. Serre.

Here are some more references:
P. Diaconis, Group representations in Probability and Statistics
B. Simon, Representations of finite and compact groups
W. Fulton and J. Harris, Representation theory: a first course

Presentations

Most of lectures in this course will be presented by students on the material in the textbook. The lectures will be distributed early in the course, and the lecture material will be assigned to you at least a week in advance. You may practice your lecture before your presentation in class.

Problems: If you can't make it for your lecture, it is important to contact me ASAP!

Homeworks

There will be 5 or 6 homeworks during the semester. The problem sets and the their due dates will be posted in the daily announcement (see below).
The lowest grade HW will be dropped. However, no late HW will be accepted .
I would not require to type your HW. However, your handwriting should be legible, it should be stapled and the edges of the HW paper should be straight.

Writing Projects

There will be one writing project. This is an expository paper (10 pages or so), written in TeX. I would encourage you to choose your own topic to write about. However, there will be a list of topics that you may choose from. It will be posted here by the end of February. Here are important due dates:
March 31 abstract
April 18 Preliminary version
May 14 final paper

Grades

Presentation 40%; attendence, class participation 20%; HW 20%; final paper 20%
There will be no exams.

TAs

Dorian Croitoru (dorian@math.mit.edu)

In case you need some more help, you can make an appointment with him. He will have an office hour from time to time.

Lecture schedule and daily announcement

Date	Lecture		Announcement
W Feb 6	Julee Kim	groups, 1.1.
F Feb 8	Alexis Brownell	1.1, 1.2
M Feb 11	Joseph Cooper	1.3, 1.4
W Feb 13	Gabriel Durazo	1.5, 1.6
F Feb 15	JK	misc	Homework 1 (due Friday February 29)
T Feb 19	Pamela Luna	2.1	A note on tensor products
W Feb 20	Janet Pan	-2.2
F Feb 22	Alejandro Palma	2.3
M Feb 25	Nevada Sanchez	2.4-2.5
W Feb 27	Yufei Zhao	2.6-2.7
F Feb 29	JK	misc.	topics for writing project (posted)
M Mar 3	Alexis Brownell	3.1-3.2	Homework 2 (due Friday March 21)
W Mar 5	Joseph Cooper	3.3
F Mar 7	Gabe Durazo	3.3, 5.7
M Mar 10	Pamela Luna	S_4	note 1
W Mar 12	Janet Pan	S_4	note 2
F Mar 14	JK	Rings and Modules
M Mar 17	Alejandro Palma	6.2-6.3
W Mar 19	Nevada Sanchez	6.4-6.5
F Mar 21	Yufei Zhao	7.1-7.2
Mar 24-28	Spring Break
M Mar 31	Alexis Brownell	7.3-7.4
W Apr 2	JK
F Apr 4	Joseph Cooper	Lie algebras
M Apr 7	Gabriel Durazo	Section B of Diaconis	Related article by Aldous and Diaconis
W Apr 9	Janet Pan	Section C of Diaconis
F Apr 11	Alejandro Palma	Section D
M Apr 14	Nevada Sanchez	Section D	Lecture note by Nevada Sanchez
W Apr 16	Alexis Brownell	Engel's, Lie's theorem
F Apr 18	Yufei Zhao	sl(2,C)
M Apr 21	Patriots day
W Apr 23
F Apr 25
M Apr 28	Yufei Zhao
W Apr 30	Janet Pan
F May 2	Nevada Sanchez
M May 5	Joy Cooper
W May 7	Alejandro Palma
F May 9	Gabe Durazo
M May 12	Alexis Brownell
W May 14