MATH 745: Lie Groups II (Spring 2000)
Title: Lie Groups II
Course web site: http://www.math.umd.edu/~jmr/745/
Meeting times: MWF, 11:00am-11:50am (MTH 2300)
Instructor: Professor
Jonathan
Rosenberg. His office is room 2114 of the Math Building,
phone extension 55166, or you can contact him by
email.
Office hours are Mondays 2-3 and Tuesdays 2-3,
or by appointment.
Text:
None. However, references will be given for various topics, and you
might want to consult a few of the following books:
- J. Frank Adams, Lectures on Lie groups
- T. Bröcker and T. tom Dieck, Representations of compact
Lie groups
- J. E. Humphreys, Introduction to Lie algebras and
representation theory
- G. P. Hochschild, The structure of Lie groups
- A. Knapp, Lie Groups: Beyond an Introduction
- V. S. Varadarajan, Lie groups, Lie algebras, and their
representations
Prerequisite: An introduction to Lie groups (MATH 744 or
equivalent)
Catalog description: A continuation of
Lie groups I in which some of the following topics will be
emphasized: solvable Lie groups, compact Lie
groups, classifications of semi-simple Lie groups,
representation theory, homogeneous spaces.
Course Description:
In the beginning God created the simple
Lie groups ...
This course will consider more advanced topics in Lie group theory
from an "interdisciplinary" point of view. In other words, I will
try to blend techniques and ideas from algebra, analysis, geometry,
and topology. Emphasis will be on some of the remarkable facts about
compact Lie groups, such as:
- The fact that the fundamental group of any compact semisimple Lie
group is finite, and a purely algebraic way to compute the
possibilities for it.
- The classification of compact Lie groups and symmetric spaces.
- The calculation of the cohomology rings of compact Lie groups and
their homogeneous spaces by purely algebraic methods.
- Vanishing of the second homotopy group of any Lie group.
- The fact that any compact Lie group is "rationally" a product of
odd spheres.
- The fact that a quotient space of a connected compact Lie group by a
closed subgroup has non-vanishing Euler characteristic if and
only if the subgroup has maximal rank.
- The Weyl integration formula and Weyl character formula.
- The Borel-Weil-Bott Theorem.
It will not be possible to cover all of these topics, but we will
do as many as time permits, depending on the interests of the class.
Course Requirements:
Homework will be assigned, collected and graded regularly,
but there will be no exams.