Spring 2007 : MAT 595 Introduction to Morse Theory

Tuesday 10:30-11:50 in 447 Altgeld.


Overview of the course

Morse theory is the study of the relation between the functions on a space and its topology.
It is an extremely powerful tool which plays an important role in many areas of geometry and topology.
Some applications of Morse theory include; Smale's proof of the Poincare conjecture in dimensions greater than four,
the Bott periodicity theorem, and theorems on the existence of closed geodesics.

In this course we will first discuss the basic machinery of Morse theory starting with the material described in Milnor's classic text.
We will also study Morse-Bott theory, and the Morse theory of manifolds with boundary.
We will then discuss the modern formulation of these ideas due to Thom, Smale, Witten and Floer.
This goes under the name of Morse homology, and is a finite-dimensional model of Floer homology.

The remainder of the class will be devoted to applications of these tools.
These will be chosen according to the tastes of the participants and are subject to the limitations of the instructor.
They may include; the existence of closed geodesics, the Bott periodicity theorem, the Morse theory of moment maps
in symplectic geometry, and the Morse-Novikov theory of closed one--forms.

Course grades will be based on participation (50%) and on 4-5 Homework assignments (50%).


Prerequisites

The prerequisites for this class are basic differential topology and algebraic topology at the level of
Guillemin and Pollack's book "Differential Topology" and Vassiliev's book "Introduction to Topology."
If you have taken Math 520 you should be well equipped for this class.


Office Hours

Tuesdays and Thursdays 1:00 to 2:00, or by appointment.


References

"Morse Theory" by J.W. Milnor, Annals of Math. Studies, vol. 51, Princeton University Press, 1963.


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This page last modified by Ely Kerman
Friday, 20-Jan-2005 13:12:53 EST
Email corrections and comments to ekerman@math.uiuc.edu