- Dani Alvarez-Gavela: Wrinkled Lagrangian Embeddings (simplyfying the singularities of Lagrangian fronts)
Melanie Bertelson: An introduction to h-principles/A correspondence between contact forms and a class of cone structures (2 talks)
First talk: An introduction to h-principles
This talk is a first course on Gromov's theory of h-principles. Its aim is to introduce the terminology and convince the audience of the validity of the h-principle on open manifolds for open invariant relations, including notably the immersion, submersion, symplectic and contact relations, through the Eliashberg-Mishachev holonomic approximation technique.
Second talk : A correspondence between contact forms and a class of cone structures
In his paper Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Sullivan shows that existence of a symplectic structure on a closed manifold is equivalent to that of an ample cone structure of bivectors that satisfies a certain homological condition. As will be explained in the talk, this equivalence can be adapted to the contact case.
Kai Cieliebak: The topology of rationally and polynomially convex domains
Abstract: Rationally and polynomially convex domains in $\C^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.
Yasha Eliashberg: TBA (3 talks)
Alexander Kupers: Homological stability and the stable homology of mapping class groups (2 talks)
Abstract: (1) Homological stability for mapping class groups.
We will explain a proof of Harer's theorem on homological stability for mapping class groups of surfaces, stressing the general method over the details of this particular case. We will then reformulate this to a geometric statement about subbundles of surface bundles up to bordism that is used in the next lecture.
(2) The stable homology of mapping class groups.
After having seen that the homology of mapping class groups stabilizes, one of course wants to compute the stable homology. The Madsen-Weiss theorem says it given by the homology of an infinite loop space of a certain Thom spectrum. One can reformulate this result geometrically as an h-principle for surface bundles with stable ends up to bordism, which Eliashberg-Galatius-Mishachev proved using wrinkling and homological stability.
Francois Laudenbach: Gromov's h-principle for contact structures
from the point of view of Haefliger structures (2 talks)
Gael Meigniez: Thurston's methods in the h-principle for foliations, and what more we can do with it (2 talks)
Yoshi Mitsumatsu: A proof of Thurston's h-principle for 2-dimensional foliations
Abstract: We recreate an unpublished proof of Thurston's h-principle
for 2-dimensional foliations which says that any smooth 2-plane field
on a manifold of dimension at least 4 is homotopic to the tangent plane
field of a foliation. We learned a simplest proof for 4 and 5
dimensional case from Andre Haefliger via Takashi Tsuboi and it is completed
along its line by using a result by Haller-Rybicki-Teichmann. As an issue, the (q+1)-connectivity for codimension-q Γ-structures with trivialized normal bundle is concluded without passing through the Mather-Thurston theory but
only by using Haefliger's original argument used to prove q-connectivity.
This is joint work with Elmar Vogt.
Emmy Murphy: Wrinkled embeddings and loose Legendrians/An alternative proof of Lagrangian caps using wrinkling techniques (2 talks)
First Talk: Wrinkled embeddings and loose Legendrians
In contact manifolds of dimension at least 5, there is a natural class of Legendrians, called loose Legendrians, which are classified due to their h-principle properties. The reason behind this is the similarity of Legendrian fronts and wrinkled embeddings. This will lead us to study "wrinkled Legendrians", an intermediate object, and see how a geometric hypothesis allows us to smooth singularities. We will also discuss some more recent results, connecting loose Legendrians with overtwistedness and open book decompositions.
Second Talk: An alternative proof of Lagrangian caps using wrinkling techniques
The existence theorem for Lagrangian caps, originally joint work with Eliashberg, essentially states than a loose Legendrian can be made into the concave boundary of an exact Lagrangian whenever it is topologically plausible. The original proof relies on performing a "Whitney trick", canceling self-intersection points of a Lagrangian immersion using graphs of Legendrian isotopies. We present an alternative unpublished proof. This proof does not explicitly rely on the classification theorem of loose Legendrians, rather it uses the definition of looseness to explicitly build Lagrangian caps by applying wrinkling techniques to the front projection of the Lagrangian cap itself.
There will be a further talk entitled "Questions in symplectic and contact flexibility" in the departmental Colloquium on Thursday 16:30-17:30.