Format

We will give a two times two hours per week course without official excercise sheets and sessions. Nevertheless we will try to include discussions of examples and excersises informally whenever such come up. Instead of a final exam we want to hand out little projects over the semester break and have an oral exam at the end of the semester break, where you mainly present your project.

    Time: Tuesday 2-4 pm & Thursday, 4-6 pm
    Room: Tue, SR 9 & Thu, SR 8
    Language: English
    Credits: 6 CP

Prerequisites

The course is aimed at math master and advanced bachelor students. Basic knowledge of differential geometry and topology (i.e. manifolds, vector fields, differential forms, ...) and also bits of Riemannian geometry will be helpful. It will not be necessary to know anything about (co-)homology, but certainly useful.

Plan of the Lecture

    The lecture will be split into four parts, every part will be held by a different lecturer.

Classical Morse Theory (Johanna Bimmermann)

    We start of with classical Morse theory. This means we discover how critical points of a generic smooth function on a closed manifolds encodes its topology. We then generalize to Morse-Bott theory (i.e. allow functions with critical submanifolds) and as application discuss homotopy groups of spheres. If time permits we could have a look at moment maps for Hamiltonian group actions on symplectic manifolds, which give a particular nice class of Morse-Bott functions.

Morse Theory on Hilbert Manifolds (Valerio Assenza)

    The second part of the course is devoted to understand how a lot of Geometrical problem can be translated in a Variational approach. Involving the Morse Theory of Critical Point we go through some of the “classical” theorem which describe relations between Topology of a space and its Riemannian structure. Below you find the schedule which with a non-zero probability ill be covered:
    • Riemannian Manifold and Curvature
    • Varitaions, Jacobi Fields and Conjugate Points
    • Energy Functional: first and second Variation
    • Bonnet-Myers and Weinstein Theorems
    • Morse Index Theorem
    • Mountain Pass Theorem and existence of Periodic Geodesic on Closed Riemannian Manifold

Morse Homology (Levin Maier)

    In the third part of the lecture we want to use Morse theory to construct a topological invariant called Morse homology of a closed Riemannian manifold. For construction we choose a Morse-Smale pair, later we see that this choice is generic, and construct using the negative gradient flow a chain complex, the Morse-Smale-Witten complex. It turns out that the homology of the Morse-Smale-Witten complex, the Morse homology, is independent of the choice of the Morse-Smale pair. Furthermore we will show that the Morse homology is isomorphic to the singular homology of the manifold. So we can say that Morse homology is a dynamical/ analytical approach to homology. Afterwards we will discuss some examples. If the time permits we could have a look at the Morse homology of the loop space.

Towards Floer Theory (Dr. Richard Siefring)

    In the final part of the course we will give an overview of Floer homology and its applications in symplectic topology and Hamiltonian dynamics. Here the tools of “classical” infinite dimensional Morse theory break down because the indices of critical points are infinite. However, by replacing the gradient flow equation with an elliptic PDE that formally has many of the same properties, one can still make sense of the “index difference” along a trajectory, and obtain a theory that works in cases where the classical theory does not.

References

[1] J. Milnor; Morse Theory
[2] R. L. Cohen, K. Iga, P. Norbury; Topics in Morse Theory
[3] A. C. da Silva; Lectures on Symplectic Geometry
[4] D. McDuff & D. Salamon; Introduction to Symplectic Topology
[5] R. Bott; The Stable Homotopy of the Classical Groups
[6] M. Audin; Morse Theory and Floer Homology
[7] D. Salamon; Lecture on Floer Homology
[8] A. Floer; A Relative Morse Index for the Symplectic Action
[9] A. Floer; Witten's Complex and Infonote-Dimensional Morse Theory
[10] A. Floer; Symplectic Fixed Points and Holomorphic Spheres