### Katastrophentheorie/Catastrophe Theory

### (Singularitäten differenzierbarer Abbildungen)

#### WS23/24, Mi 12-14, M103

#### Seminar Description

The Seminar is concerned with the study of the

*local behavior* of real-valued smooth functions of several variables and

the

*classification problem* for the types of singularities that can arise at the critial points.

We will develop methods to study these singularities through the properties of their Taylor polynomials and investigate

their genericity and stability properties.

Moreover, we will introduce the fundamental notion of

*(universal) unfolding* that involves smooth functions with additional

parameters and examine the dependence of the critical sets on the parameters.

A

*catastrophe* (in the mathematical sense used in the theory) occurs when the critical set

*changes its form suddenly* under

a small pertubation of the parameters. Catastrophe Theory, introduced by René Thom, develops a fascinating viewpoint on

the qualitative understanding of singularities of smooth functions with many applications in other scientific fields.

** See the Informal Overview for
more details.**
The topic of the Seminar forms a natural continuation of the lecture courses in Analysis but will also make significant use of

methods from the lecture courses in (Linear) Algebra. The Seminar should be of interest to students at any level interested in

Analysis or (Differential) Geometry and Topology or Mathematical Physics.

The Seminar talks will be held in German or English. The written reports can be submitted in German or English.

** Please contact me by e-mail if you are planning to attend the seminar.**
** Seminar Schedule**: See

here.

#### Recommended Reading for the Seminar

Th. Bröcker, ''Differentiable Germs and Catastrophes''

D.P.L. Castrigiano, S.A. Hayes, ''Catastrophe Theory''

Y.-C. Lu, ''Singularity theory and an Introduction to Catastrophe Theory''

T. Poston, I. Stewart, "Catastrophe Theory and Its Applications".

Also see

here for material from a related seminar which
might be useful.

#### Prerequisites

Analysis I-II/Lineare Algebra I-II/(some) Algebra