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.