Organizer: Karen Uhlenbeck
Tu-Th RLM 9.160
This course is a seminar course which is intended for first and second
year graduate students. We will cover some topic which is
connected to geometry. In particular, the first year RTG students
in geometry are expected to take this seminar. Students who are
officially in other subjects or who are undecided are very welcome.
The course is in seminar format. Students will present most, if
not all, of the material. Usually there is a restriction on the number
of students allowed to register, so that every student will present
material at least three times during the course of the semester.
This keeps the students in the seminar on their toes for the
term! There is no need to be particularly knowledgeable or expert
in the field. The prerequisites will, to some extent, depend on
the choice of topic. Part of the point of the seminar is to learn to
read papers without knowing all the background.
The two topics which have been covered in previous
seminars of this sort are:
Morse Theory (seminars
in previous years used a
text by John Milnor): Here some basic knowledge
of algebraic topology as in the prelim
course is advisable. This is very basic material in geometry and
topology. Yes, we will use some ODE, but few students have much of a
background in this field. It is a good place to learn.
(various papers): For this topic it might be helpful to have had a
course on curves and surfaces, but most of the participants did
not. Also at least attending the differential topology prelim
course concurrently would be helpful. This topic is aimed at attuning
students towards the ideas involved in the solution of the Poincare
conjecture using Ricci flow. Yes, we use some PDE, but there
is no prerequisite of this sort.
The topic for Spring 2010 will be!
Semi-Simple Lie Algebras and Compact Lie Groups
Reading material: Notes on Lie Algebras by Hans Samuelson
This is out of print, but available in compact version used and as a
big pile of paper free, both off the web.
The prerequisites are a solid knowledge of linear algebra and group
theory and some mathematical sophistication. This classification
is one of the beautiful acheivements of mathematics during the first
half of the 20th century.