## RET Summer School on Distributions and h-Principles

h-Principles are a powerful tool that lie on the interface between the geometry of manifolds and algebraic topology. They relate moduli spaces of geometric structures to certain spaces of sections of fibre bundles, and they are often able to establish a weak homotopy equivalence between them, in which case the geometric structure is said to be flexible (otherwise it is called rigid).

The geometric structures considered in this course will be regular distributions in the tangent space of a manifold, that is, sections of the fibrewise Grassmannian of this tangent space that are solutions of a partial differential relation that do not possess local invariants. Therefore the moduli problem is purely global on the manifold.

In this course we will study three examples of regular distributions (foliations, contact structures, and Engel structures), where there is a dichotomy between flexible phenomena and rigid phenomena. These lectures will explain old and (mostly) recent developments in the flexible side, but continuous references to its rigid nemesis will be provided. Hence the three theories can be understood as a struggle between the two sides.

This summer school is part of a series of activities organised by the Spanish Topology Network (RET).

Program:

- Introduction to the h-Principle
- Contact Flexibility
- Engel Flexibility
- Foliation Flexibility
- Optional: Donaldson Divisors

Recommended readings:

- Y. Eliashberg and N. Mishachev,
*Introduction to the h-Principle*, Graduate Studies in Mathematics 48, Amer. Math. Soc., Providence, 2002.

Further readings:

- M. Gromov,
*Partial Differential Relations*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1986. - Y. Eliashberg,
*Classification of overtwisted contact structures on 3-manifolds*, Invent. Math. 98 (1989), no. 3, 623-637. - M. S. Borman, Y. Eliashberg and E. Murphy
*, Existence and classification of overtwisted contact structures in all dimensions*, Acta Mathematica 215 (2015), no. 2, 281–361. - Y. Eliashberg and N. M. Mishachev,
*Wrinkling of smooth mappings and its applications I*, Invent. Math. 130 (1997), 345-369. - Y. Eliashberg and N. M. Mishachev,
*Wrinkling of smooth mappings and its applications II. Foliations of codimension greater than one*, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 321–350. - Y. Eliashberg and N. M. Mishachev,
*Wrinkling of smooth mappings and its applications III. Wrinkling of embeddings and K. Igusa's theorem*, Topology 39 (2000), no. 4, 711–732.

Lecturers:

- Roger Casals, MIT
- Álvaro del Pino, ICMAT
- Francisco Presas, ICMAT

Organisers: Federico Cantero, Carles Casacuberta, Eva Miranda and Ignasi Mundet i Riera.