# Mini-school on Cerf theory and the Pseudoisotopy Problem

Presenter: Filippos Sytilidis (Oxford, UK)

Flyer for course

Each lecture is 2 hours.

## Course description

The classical pseudo-isotopy problem asks when a pseudo-isotopy (i.e. a generalised isotopy that is not required to be level-preserving) is isotopic to an isotopy. This problem has deep connections with many areas of topology, from the singularity theory of smooth mappings to algebraic K theory, as well as with gravitational lensing in astronomy. We will introduce the theory of Morse functions and their deformations (Morse-Cerf theory) and discuss how Cerf used it to address some cases of the pseudo-isotopy problem in his seminal work.
This mini-school will be of interest to topologists and mathematical physicists. Please register via both links above to attend all the lectures.

Please register to attend Wednesday lectures and Thursday lectures.

## Lecture schedule

Lecture 1: Wed 6 September, Introduction to pseudo-isotopy and Morse theory, Youtube, Notes

Lecture 2: Wed 13 September, Morse theory and Thom’s Jet Transversality Theorem, Youtube, Notes

Lecture 3: Wed 20 September, Thom’s Jet Transverality Theorem II Youtube, Notes

Lecture 4: Thu 21 September, An Introduction to Cerf Paths Youtube, Notes

Lecture 5: Wed 4 October, Ordering of Critical Points Teams Link

Lecture 6: Wed 11 October, Overview of the Last Step in Cerf’s Proof Teams Link

## Preparatory material

- For Lecture 2: Guillemin and Golubitsky (G&G), Stable mappings and their singularities, Chapter 2 (Transversality), sections 1-4.
- For Lectures 3-4: Chapter VI, section 2 of (G&G)
- For jet bundles: Michor, Manifolds of differentiable mappings, chapter 1.