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 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:
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.
Thursday 17 January, 2013 16:00-17:30 UCL maths, Room 807 |
Introduction (jet bundles and PDRs, flavors of the h-principle, sphere eversion and other sample applications). Speaker: Chris Wendl (UCL) scanned lecture notes |
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Thursday 24 January, 2013 | no seminar |
Thursday 31 January, 2013 16:00-17:30 UCL maths, Room 807 |
The holonomic approximation theorem. Speaker: Jason Lotay (UCL) scanned notes (by Chris) |
Thursday 7 February, 2013 16:00-17:30 UCL maths, Room 807 |
The h-principle for open Diff-invariant relations and applications. Speaker: Johannes Nordström (Imperial) scanned notes (by Chris) |
Thursday 14 February, 2013 16:00-17:30 UCL maths, Room 807 |
Convex integration and ample differential relations Speaker: Jonny Evans (UCL) cartoon preview and lecture notes (on Jonny's blog) |
Thursday 21 February, 2013 16:00-17:30 UCL maths, Room 807 |
Microflexibility, local integrability and capacious groups Speaker: András Juhász (Imperial) scanned notes |
Thursday 28 February, 2013 16:00-17:30 UCL maths, Room 807 |
Legendrian and Lagrangian immersions Speaker: András Juhász (Imperial) scanned notes Microextension and co-closed G2 structures Speaker: Johannes Nordström (Imperial) scanned notes |
Thursday 7 March, 2013 16:00-17:30 UCL maths, Room 807 |
Survey on Stein manifolds Speaker: Chris Wendl (UCL) scanned notes (see also Kai Cieliebak's lecture notes from the CAST Summeer School, Budapest 2012) |
Thursday 14 March, 2013 16:00-17:30 UCL maths, Room 807 |
h-principles for the curvature of (semi-)Riemannian metrics Speaker: Marc Nardmann (Hamburg) slides from Marc Nardmann's talk |
Thursday 21 March, 2013 16:00-17:30 UCL maths, Room 807 |
Convex integration and contact geometry Speaker: Hansjörg Geiges (Cologne) scanned notes (erroneously labelled "talk 8") |
For more information about the seminar contact Chris Wendl by sending e-mail to c dot wendl at ucl dot ac dot uk