RET Summer School on
Distributions and h-Principles

To be held at the University of Barcelona in 2017, from Monday 10 July to Saturday 15 July
Lectures by Roger Casals, Álvaro del Pino and Francisco Presas

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).


  • 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.


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


  Monday Tuesday Wednesday Thursday Friday Saturday
09:30 - 10:30 Lecture 1 Lecture 5 Lecture 9 Lecture 12 Lecture 16 Lecture 20
10:30 - 11:00 Coffee break Coffee break Coffee break Coffee break Coffee break Coffee break
11:00 - 12:00 Lecture 2 Lecture 6 Lecture 10 Lecture 13 Lecture 17 Lecture 21
12:15 - 13:15 Lecture 3 Lecture 7 Lecture 11 Lecture 14 Lecture 18 Lecture 22
15:30 - 16:30 Lecture 4 Lecture 8   Lecture 15 Lecture 19  
16:30 - 17:00 Break Break   Break Break  
17:00 - 18:00 Complements Complements   Complements Complements  


To register, please fill in and submit the following registration form before May 31st, 2017. The registration fee amounts to 50 € and includes coffee breaks and a group meal. It is payable by bank transfer to the following account:

Owner: UB-IMUB
IBAN: ES38 2100 3642 1222 0009 3134
Address: CaixaBank, Gran Via 601, 08007 Barcelona

Please, include your name and specify "hprinciples" in the transfer order.

We can offer financial support to cover stays of young researchers. For that, we ask you to send us your CV together with a letter of recommendation to before April 30th.


In this map you can find the university residences and a couple of closeby hotels.