Morse theory tutorial - Fall 2003

Time and location:

Wednesdays 4-6pm, SC 116

Instructors:

Erick Matsen

Email: matsen_at_math.harvard.edu

Office: Science Center 428c

Office hours: Tuesday 3:30-4:30 and by appointment

Ciprian Manolescu

Email: manolesc_at_fas.harvard.edu

Office: Science Center 431f

Office hours: Monday 3-4 pm and by appointment

Homepage:

http://www.people.fas.harvard.edu/~manolesc/morse.html

Grading:

30% homework, 20% oral presentation, 50% final paper.

Homework:

Problem set 1 is here.

Problem set 2 is here.

Problem set 3 is here.

Problem set 4 is here.

Problem set 5 is here.

Outline:

Morse theory is an extremely simple tool which has revolutionized fields of mathematics several times over. Morse himself developed the theory and applied it to mathematical physics. Later, Bott took these ideas and used them to prove his celebrated periodicity theorem. Then Smale used it to prove the h-cobordism theorem, which implies the generalized Poincare conjecture in dimensions five and above. More recently Andreas Floer applied the ideas in the symplectic setting to prove the Arnol'd conjecture, and in the process invented an important new homology theory.

This tutorial, however, will have the goal of introducing the basic ideas and proving Bott's periodicity theorem. This theorem is concerned with the topology of matrix groups, and demonstrates a very beautiful and suprising fact about their higher homotopy groups. In a larger sense, though, another goal for the course will be to play with manifolds and the tools that we use to understand them. We will get to talk about Lie groups and differential geometry (two extremely important parts of mathematics) in very elementary, hands-on ways. It should be a lot of fun!

Prerequisites:

Knowledge of manifolds at the level of Math 134 or 135.

Topics to be covered:

Morse functions, manifolds as CW complexes, the Morse inequalities;
Riemannian geometry: first and second fundamental forms, connections, curvature;
Morse theory on loop spaces with the energy functional;
Lie algebras, Lie groups and symmetric spaces;
homotopy groups;
Bott periodicity theorem for the unitary group.

Suggestions for project topics:

Morse theory and Heegaard splittings;
Morse homology;
Bott periodicity for the orthogonal group;
Morse-Bott functions;
Picard-Lefschetz theory;
Topology of algebaric varieties;
The h-cobordism theorem;
Ljusternik-Schnirelmann theory;
Catastrophe theory;
The Conley index and applications.

Textbook:

J. Milnor, "Morse theory," Princeton University Press, Princeton, 1963.