Seminar: Katastrophentheorie / Catastrophe Theory, Sommersemester 2024

General information

Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage)

Moodle: Everyone who attends the seminar should join the moodle at https://moodle.hu-berlin.de/course/view.php?id=126925 in order to receive occasional time-sensitive announcements. The enrollment key is: keinePanik
(Note: If you do not have a HU computer account, it is also possible to join the moodle using a non-HU e-mail address, it just requires a few extra clicks.)

Time and place: Fridays 9:15-10:45 in room 3.008 (Rudower Chaussee 25)

Language: Die vorhandene Literatur für das Thema des Seminars ist (soweit mir bekannt) alles auf Englisch, aber für ihre Vorträge dürfen Vortragende im Seminar frei zwischen Deutsch und Englisch wählen. / The available literature for the topic of the seminar is (as far as I know) all in English, but speakers in the seminar may choose freely between German and English for their talks.

Prerequisites: The HU courses Analysis I-III, Lineare Algebra I-II and Algebra und Funktionentheorie, or equivalent. More precisely, participants will need to have a solid understanding of differential calculus for functions of several real variables (including most crucially the implicit function theorem), the basic existence/uniqueness theory for ordinary differential equations (as covered in Analysis III), and some basic knowledge of rings, ideals and modules. Further knowledge of commutative algebra is not required, as the necessary tools (e.g. Nakayama's lemma) will be introduced in the course of the seminar. Knowledge of the theory of smooth manifolds is not necessary, but may occasionally be helpful; we will at least frequently mention the notion of submanifolds of Euclidean space and their tangent spaces, as arise for instance from standard applications of the implicit function theorem.

The subject: Catastrophe theory is a slightly sensationalistic term for a branch of pure mathematics that can be used to model sudden or violent (i.e. discontinuous) changes in systems that depend smoothly on external parameters. The subject earned a mildly dubious reputation during the 1960's and 70's, due to a certain amount of overzealous hype about its applications to the natural sciences and humanities. But as a purely mathematical discipline, catastrophe theory is an elegant synthesis of differential calculus with commutative algebra, one that furnishes answers to many important questions arising in the study of dynamical systems, differential geometry, topology, and other areas of both pure and applied mathematics. Catastrophe theory is also a special case of -- and an accessible entry point into -- the larger subjects of singularity theory and bifurcation theory, which study the qualitative structure of smooth maps (and their dependence on extra parameters) near points at which the usual hypotheses of the inverse and implicit function theorems fail. In this seminar, we will mostly not discuss applications, but focus instead on the mathematical underpinnings of catastrophe and singularity theory. Our first major goal will be to understand a famous result of René Thom, which classifies the qualitative structure of all possible catastrophes for systems depending on at most four parameters: in essence, every such system that can arise in practice matches one of seven explicit local models, known as the seven elementary catastrophes. After this, we will have some time to discuss how the ideas behind that theorem generalize into the wider contexts of singularity and bifurcation theory.

See this seminar plan for a more detailed summary of what the subject is about and a week-by-week outline of topics.

Requirements: All (students and otherwise) are welcome to attend the seminar and may volunteer to give talks, though priority will be given to Bachelor- and Master-students (in that order) who need credit for the seminar. For students to receive credit, the requirements are the following:

1. Give at least one of the talks (normally this will mean one 90-minute session, but these sessions could be split up if there are too many volunteers), with careful attention to fitting all necessary material within the given time constraints;
2. Submit clearly readable notes for your talk (you are welcome to TeX them, but handwritten notes are also fine), so that they can be scanned and uploaded to this website by the following Monday;
3. Attend the seminar regularly (at most three absences in the semester, barring exceptional circumstances).

Before you volunteer to give one of the talks, you should look at the detailed seminar plan for more information and suggestions about each of the topics. Since we expect to have a mixture of Bachelor- and Master-students, I have marked with an asterisk^* the topics for which I'd suggest a Master student give the talk, but this is not a rule set in stone, and one or two of them are probably borderline cases.

UPDATE 19.04 (after the first meeting): Most topics have now been assigned, but the ones marked "TBA" on the schedule below are still available; the June 7 talk on plane curve singularities is also officially still available, as Gerard has stated that he will step back if any Bachelor student really wants to take that one. If you'd like to volunteer for any of these, send me an e-mail! (If there are too many volunteers, we can also discuss splitting some talks between two people.)