Many natural problems in differential geometry and topology -- for example the existence of immersions, symplectic forms, or isometric maps -- can be formulated in terms of partial differential relations (PDRs), i.e. equations or inequalities that constrain the partial derivatives of a map. In solving such problems, there is typically a much easier problem that must be solved first: the classification of formal solutions, maps which satisfy a corresponding algebraic relation without any constraints on derivatives. Understanding formal solutions is typically a matter of standard homotopy theory -- it may in general by very easy or very hard, but on the surface it is at least much simpler than solving a PDE or PDR. The remarkable discovery, which originated with various seemingly unrelated results in the 1950's and was then formalized by Gromov around 1970, was that for certain classes of PDRs, finding formal solutions is not only necessary but also sufficient: one says that a problem "satisfies the h-principle" (the "h" here stands for "homotopy") if every formal solution is homotopic to a genuine solution, or even better, if the space of genuine solutions is homotopy equivalent to the space of formal solutions. In this case the problem is described as "flexible," meaning essentially that it can be solved by purely homotopy theoretic methods, with no need for deeper analytical or geometric techniques.

The h-principle comes in a variety of flavors and has applications in wide range of subjects. Famous examples include the following:

• The Smale-Hirsch immersion theory in differential topology
• The Nash-Kuiper C¹-isometric embedding theory in Riemannian geometry
• Existence and uniqueness (up to deformation) of symplectic and contact structures on open manifolds

The goal of this seminar will be to learn the basics of the subject, including at least the three classic applications listed above, and two powerful methods for proving fairly general h-principles: holonomic approximation and convex integration theory. We will mainly follow the book by Eliashberg and Mishachev (first item on the literature list below). Additional topics will depend on the interests of the participants and may include any of the following:

• more advanced applications in symplectic and contact geometry
• applications to foliation theory
• metrics with negative curvature
• further topics to be discussed

We assume the target audience for this seminar to be familiar with the basic notions of differential geometry and algebraic topology, including vector and fiber bundles, cohomology and the standard characteristic classes. A healthy interest in symplectic geometry also could not hurt, but prior knowledge of symplectic geometry will not be assumed.

We plan to meet normally every week on Thursdays at 16:00 in Room 807 of the UCL mathematics department, located at 25 Gordon Street in Bloomsbury (very close to Euston Station). Sessions will typically last between 60 and 90 minutes.

• Y. Eliashberg and N. Mishachev, Introduction to the h-principle. Graduate Studies in Mathematics 48. AMS, Providence, RI, 2002.
  This is a very good book for learning the essentials of the subject, with particularly good coverage of the applications to symplectic geometry. In addition to a detailed discussion of convex integration, it introduces the method of holonomic approximation, which simplifies the proofs of results that previously required the covering homotopy method.
• H. Geiges, h-principles and flexibility in geometry. Mem. Amer. Math. Soc. 164 (2003), no. 779.
  A very short book that provides an introduction to the basic ideas, focusing on the covering homotopy method and its applications, and a sketch of convex integration.
• M. Gromov, Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986.
  The bible of the subject, but a difficult read for beginners.
• Lecture notes on convex integration by Vincent Borelli, from the September 2012 workshop in Les Diablerets:
  1. One dimensional convex integration
  2. The h-principle for ample relations
  3. The h-principle for Isometric Embeddings
  4. Flat 2-tori in three dimensional space
  Borelli has also written a nice short survey article on the h-principle in French.
• h-principles for negative Ricci and scalar curvature, slides from a talk by Marc Nardmann at the September 2012 workshop in Les Diablerets (go to this page and click on the relevant link under "Information")
• S. Tabachnikov, A cone eversion. Amer. Math. Monthly 102 (1995), 52-56.
• See also the link below to John Francis's course The h-principle in topology, which includes lecture notes

• Outside In: a very entertaining and informative 20-minute video by The Geometry Center explaining Thurston's construction of the Smale sphere eversion

• The h-principle Seminar and Autumn school, a learning seminar in Bern (Spring 2012) followed by a workshop in Les Diablerets (September 2012)
• Gromov's h-principle and its applications, a research seminar run by Andrés Angel and Johannes Ebert at the Universität Bonn (Summer semester 2010)
• The h-principle in topology, a course by John Francis at Northwestern University (Spring 2011); includes lecture notes!

For more information about the seminar contact Chris Wendl by sending e-mail to c dot wendl at ucl dot ac dot uk