S2D1/S4D1 - Hauptseminar Geometrie / Graduate Seminar on Differential Geometry

Seminar Title: Morse theory with a view towards Floer theory

• Instructor: Laurent Côté
• Email: lcote@math.uni-bonn.de

Practical information:

• Preliminary meeting: Thursday Feb 13 at 4pm; N 0.003 (Neubau). Registration for the seminar will take place during the preliminary meeting. In case there are not enough available slots, students who are writing a Bachelors or Masters thesis with me (or with someone else on a related topic) will get priority.
• Seminar location: Endenicher Allee 60 - SemR 0.003
• Seminar time: Fridays 12-14. Talks will start at 12:15 and last 90 minutes, including a five minute break during the talk.

Overview of the seminar:

A smooth real-valued function on a smooth manifold is said to be Morse if all of its critical points are non-degenerate. In other words, the ''Hessian matrix'' of second derivatives must be non-singular at all critical points. It is an important fact that Morse functions exist on any smooth manifold.

Morse functions have the virtue that one can precisely describe how the topology of the sub-level sets changes when one passes a critical point. This leads to a rich theory relating the topology of smooth manifolds with the study of Morse functions on them. This theory was pioneered by M. Morse in the mid twentieth century and was arguably the key tool in the spectacular development of differential topology in the 1950s and 60s due to Bott, Smale, Milnor and many others.

There is another approach to Morse theory which is analytical rather than topological. Its roots are murky to the instructor, but it was at least partly popularized by Witten in the early 1980s, for reasons having to do with high energy physics.

This analytical approach to Morse theory does not yield any new applications in topology. However, in the mid 1980s, Andreas Floer initiated an infinite-dimensional generalization which we now refer to as Floer theory (or Morse-Floer theory). Floer theory produces a package of invariants which are of central interest in many parts of geometry and topology, particularly in differential topology and symplectic geometry. The development of Floer theory has changed the face of many areas of mathematics, and continues to be a central topic today.

While Floer theory is analytically much deeper than Morse theory, both theories share the same formal properties. A solid grasp of the analytical approach to Morse theory is a huge help, and arguably a pre-requisite, for studying Floer theory.

The main goal of the seminar is to understand the analytical approach to Morse theory. Towards the end the semester, we will move on to discussing some basic symplectic geometry. The last talks will be about Floer theory and (some of) its applications to symplectic geometry. We will not discuss the analytical foundations of Floer theory, and focus only on the formal structure and analogies with Morse theory.

The seminar will be in English.

Pre-requisites: This seminar is suitable for both advanced Bachelors students and Masters students. Students participating in the seminar are expected to be familiar with the content of the following courses:

• Analysis and Geometry on Manifolds (V3D3/F4D1)
• Topology I (V3D1/F4D1

Plan of talks: The talks will closely follow the wonderful, and very detailed book of Audin and Damian. A detailed plan will be posted to this webpage in due course.

Some relevant literature:

• M. Audin and M. Damian, Morse theory and Floer homology Springer Universitext, 2014. (This remarkable book will be our main source for the seminar. The first part of the book gives a complete treatment of the analytical approach to Morse theory. The second part contains an introduction to symplectic geometry, and a complete treatment of Hamiltonian Floer homology in the simplest setting.)
• J. Milnor, Morse theory. Princeton university press, 1963. (This is without contest the most famous book on Morse theory, and is regularly cited as one of the best written books in the history of mathematics.)
• L. Nicolaescu, An Invitation to Morse Theory. Springer, 2011. (This is a very detailed book on Morse theory, which also discusses stratification theory and complex Morse theory.)
• M. Schwarz, Morse homology. Progress in Mathematics, 111. Birkhäuser Verlag, Basel, 1993. (This is another excellent book for students interested in the analytical foundations of Morse theory, in preparation for studying Floer theory.)