RET Summer School on Distributions and h-Principles
This is the lecture series website for the Summer School 2017 from Monday July 10 to Saturday July 15 at Universitat de Barcelona.
Here's us. Also, in 1983 S. Dalí completed the painting The Swallow's Tail, it contains the main ideas of the course.
Research articles:
Contact and Symplectic Geometry
- Classification of overtwisted contact structures on 3-manifolds, Y.M. Eliashberg, Invent. Math. (1989).
- Existence and classification of overtwisted contact structures in all dimensions, M.S. Borman, Y.M. Eliashberg, E. Murphy, Acta Math.(2015).
- Almost contact 5-manifolds are contact, R. Casals, D.M. Pancholi, F. Presas, Annals Math. (2012).
- Geometric criteria for overtwistedness, R. Casals, E. Murphy, F. Presas (2015).
- Loose Legendrian Embeddings in High Dimensional Contact Manifolds, E. Murphy (2012).
Engel structures
- Existence h-principle for Engel structures, R. Casals, J.L. Pérez, A. Pino, F. Presas, Invent. Math. (2017).
- On the classification of prolongations up to Engel homotopy, A. Pino, Proc. AMS (2017).
The Wrinkling Saga
- Wrinkling of smooth mappings and its applications I, Y.M. Eliashberg, N.M. Mishachev, Invent. Math. (1997).
- Wrinkling of smooth mappings II. Wrinkling of embeddings and K. Igusa's theorem, Y.M. Eliashberg, N.M. Mishachev, Topology (2000).
- Wrinkling of smooth mappings III. Foliations of codimension greater than one, Y.M. Eliashberg, N.M. Mishachev, Topol. Methods Nonlinear Anal. (1998).
- Topology of spaces of S-immersions, Y.M. Eliashberg, N.M. Mishachev, Progr. Math. 296 (2012)
Textbook References:
- Singularities of caustics and wave fronts, V.I. Arnol'd, Mathematics and its Applications 62 (1990).
- Partial differential relations, M. Gromov, Springer Verlag (1986).
- Introduction to the h-principle, Y.M. Eliashberg, N.M. Mishachev, AMS-GSM 48. (2002).
- Symplectic geometry, V.I. Arnol'd, A.B. Givental', Dynamical Systems IV, Encyclopaedia Math. Sci. (2001).
- An introduction to contact topology, H. Geiges, CUP (2008).
- Symplectic geometry of affine complex manifolds, K. Cieliebak, Y.M. Eliashberg, AMS (2012)
Lecture Diaries
In these diaries we record the main ideas covered in each day of the lecture series.
- h-Principles
Proof of Whitney-Grauert Theorem. Immersions.
Approximating curves with controlled derivatives.
Convex integration. Proof of Nash-Kuiper Theorem.
Partial differential relations. h-Principle Toolbox.
Test Exercises
(a) Show that the group of connected components of the space of formal immersions of a circle into the plane is the integers.
(b) Show that the group of connected components of the space of formal immersions of a 2-sphere into 3-space is Z/2Z.
(c) Approximate the parametrized vertical axis in 3-space with a parametrized curve whose derivative in the vertical direction is small
and its projection to the horizontal plane avoids a fixed line.
(d) Prove the Whitney-Grauert Theorem from the Holonomic Lemma by thickening the circles to annuli.
(e) Describe the partial differential relation on k-forms imposed by closedness.
- Contact Topology
Topologically stable distributions. Contact structures.
Examples. Hamiltonians and contactomorphisms.
Absolute h-principle. Bennequin's Theorem.
Integral submanifolds. Relative h-principle.
Test Exercises
(a) Show that the unit ball in standard contact space is contactomorphic to a ball of arbitrary radius.
(b) Prove that any smooth submanifold of codimension more than a half can be approximated by an integral submanifold.
(c) Show that there exists a unique almost contact structure on the 5-sphere.
(d) Find two homotopic contact structures in the 3-sphere with distinct image by the absolute scanning map.
(e) Find two smoothly isotopic Legendrian knots in 3-space with distinct image by the relative scanning map.
- Overtwisted disks: absolute h-principle
Transverse geometry and characteristic foliations. Contact germs along hypersurfaces.
Comparison problem in terms of contact Hamiltonians. Disorderability criterion.
Proof in the 3-dimensional case. Quantitative positivity in higher-dimensions.
Special Hamiltonians and overtwisted disk. Basic reduction scheme.
- Equivariant coverings and ε-universal
Reduction to graphical local model. Description in terms of monodromy.
The Heisenberg cocompact lattice. Finite list of regular saucers.
Self-replicating germs. Extensions for the overtwisted disk.
Proof in the higher-dimensional case. Consequences.
- Loose charts: relative h-principle
Legendrian singularities and wavefronts. Quantitative threshold for the action-width ratio.
Reduction to graphical local model. Ampleness for ε-Legendrian relation.
Legendrian wrinkled embeddings. Loose chart degeneration to wrinkle.
Creation of loose charts. The whack-a-mole argument.
- Geometric criteria I: Local-to-global Principle
Higher-dimensional overtwisted disks as thick 2-dimensional disks.
Quantitative neighborhoods. Branched covers and normal size expansion.
Open book decompositions. Large conjugated A_{k} singularities are overtwisted.
Flexible Weinstein cobordisms. Loose unknot implies overtwistedness.
- Geometric criteria II: Surgery calculus
Wavefront (±1)-handleslides. (+1)-surgery on loose is overtwisted.
Subcritical cocores are loose. Negatively stabilized open book is overtwisted.
Weinstein concordance from overtwisted to tight. h-principle for concave end.
Further examples. Legendrian lifts of Dehn twists. Problem session.