Introductory Graduate Geometry Seminar         Spring 2010

      Organizer:   Karen Uhlenbeck

       Time  2:00-3:15 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.

         Curve shortening (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!

         Classification of 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.