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

Proyecto MTM2015 -69385-REDT
Proyecto MTM2016-76453-C2-2-P

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
Lecture 1
Introduction to contact topology
Lecture 5
The wrinkling philosophy
(del Pino)
Lecture 9
Overtwisted contact structures II
Lecture 12
(del Pino)
Lecture 16
Weinstein manifolds II
Additional topics on Engel topology
  Coffee break Coffee break Coffee break Coffee break Coffee break Coffee break
Lecture 2
Introduction to the h-Principle
Lecture 6
Wrinkled maps I
(del Pino)
Lecture 10
Wrinkled embeddings I
(del Pino)
Lecture 13
Loose Legendrian embeddings I
Lecture 17
Rigid-Flexible dichotomy
Additional topics on Wrinkling
(del Pino)
Lecture 3
Holonomic approximation I
(del Pino)
Lecture 7
Overtwisted contact structures I
Lecture 11
Wrinkled embeddings II
(del Pino)
Lecture 14
Loose Legendrian embeddings II
Lecture 18
Engel structures
Additional topics on Wrinkling
(del Pino)
Lecture 4
Holonomic approximation II
(del Pino)
Lecture 8
Wrinkled maps 2
(del Pino)
Lecture 15
Weinstein manifolds I
Lecture 19
Overtwisted Engel structures
(del Pino)
  Break Break   Break Break  
Discussion on advanced topics
Discussion on advanced topics
  Discussion on advanced topics
Discussion on advanced topics


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.


Roisin Braddell Universitat Politècnica de Catalunya
Federico Cantero Morán BGSMath-UB
Carles Casacuberta Universitat de Barcelona
Carlos Arturo Cruz Rodríguez Universitat de Barcelona
Eduardo Fernández Fuertes ICMAT
Michael Harrison Pennsylvania State University
Luis Hernández Corbato ICMAT
Alvin Jin University of California at Santa Cruz
Dusan Joksimovic Utrecht University
Dmitry Korshunov National Research University HSE, Moscow
Mikel Lluvia Universitat de Barcelona
Lucía Martín ICMAT
Francisco Javier Martínez Aguinaga ICMAT
Daan Michiels University of Illinois at Urbana-Champaign
Eva Miranda Universitat Politècnica de Catalunya
Ignasi Mundet i Riera Universitat de Barcelona
Dorde Nikolic University of Belgrade
Noboru Ogawa Tokai University
Cedric Oms Universitat Politècnica de Catalunya
José Luis Pérez García ICMAT
Nicola Pia University of Cagliari
Wolfgang Pitsch Universitat Autònoma de Barcelona
Samuel Ranz Castañeda ICMAT
Carles Sáez Calvo BGSMath-UB-CRM
Mario Guillermo Shannon Nuñez Institut de Mathémathiques de Bourgogne
Álvaro Torras Casas University of Leicester
Aleksy Tralle University of Warmia and Mazury
Anne Vaugon Université Paris Sud
David  White Denison University
Ivan Yudin University of Coimbra
Filip Zivanovic University of Oxford


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