Introduction to the h-principle
In 2019, I teach a graduate course about the h-principle in
Main references are:
- Eliashberg-Mishachev's book Introduction to the h-principle
- Any lecture note from
- Gromov's book Partial differential relations (hard)
Student talks suggestions
In the following list, EM means Eliashberg-Mishchev's book. The first
three items are directly in line with the lectures. The last two are
much more ambitious, but giving an idea of how ideas from the lectures
can be complemented with other ideas would already be nice.
EM Thm 12.1.1. Holonomic approximation and symplectic geometry.
Holonomic approximation for microflexible relations
EM chapter 13 (aim for Thm 13.4.1, bonus 16.1.2). Variation on proof
of holonomic approximation.
First order differential operators
EM Chap 20. Convex integration for operators like divergence.
René Thom and the h-principle
Discuss how this paper
proves existence of sphere eversion following ideas from Thom.
Convex integration ans PDEs
Discuss this survey.
Positive isotopies of loose Legendrians
Main result from this paper.
Legendriennes lâches et plastikstufe
Theorem 1.1 from this paper.
Existence of contact structures in higher dimensions
Discuss something of the main proof of
One more reasonnable goal could be explaining existence on closed
3-manifold from this point of view. See also
this Bourbaki talk and references quoted at the end of
Topological characterization of Stein manifolds
Thm 1.5 from
(see also Cieliebak and Eliashberg book if needed).