Workshop on Contact Geometry in Dimension Three and Higher - Programme

schedule     minicourse synopses     titles/abstracts for research talks

Schedule (tentative)

 All talks will take place in the Harrie Massey Lecture Theatre on the ground floor of the UCL Mathematics Department, and coffee breaks will take place in the Wilkins North Cloisters. See the practical information page for more details. UPDATE: Some talks will not be in Harrie Massey Lecture Theatre, due to problems with the air conditioning. See the main page for up-to-date announcements.

Synopses of the minicourses
  1. Flexibility in higher-dimensional contact geometry
  2. Intersection theory of punctured holomorphic curves and applications
  3. Orderability and Rabinowitz Floer theory

Flexibility in higher-dimensional contact geometry (Patrick Massot and Emmy Murphy)

Overview:
We present recent flexibility results concerning Legendrian and Lagrangian submanifolds in high-dimensional contact and symplectic manifolds. The first half of the course will focus on the definition and classification of loose Legendrians in high-dimensional contact manifolds. The classification, which is an h-principle type result, reduces the geometric classification to algebro-topological invariants. The proof relies on previous h-principle results outside of the contact/symplectic world, namely convex integration and wrinkled embeddings. From here we discuss Lagrangian caps. Given a symplectic manifold with a concave Liouville end, we demonstrate the existence of exact Lagrangian submanifolds which are cylindrically loose in the concave end. The proof here relies strongly on the classification result of loose Legendrians: the classification gives non-trivial Legendrian isotopies, which can be graphed as Lagrangian cobordisms. Finally, we give applications of both results to the geometry of Weinstein manifolds: the classification of flexible Weinstein manifolds, the flexible Weinstein h-cobordism theorem, and the universal embedding property of flexible Weinstein domains. Notes:
 Main references:

Intersection theory of punctured holomorphic curves and applications (Richard Siefring and Chris Wendl)

Overview:
Most applications of pseudoholomorphic curves to symplectic 4-manifolds have depended heavily on positivity of intersections and the adjunction formula, which give homotopy-invariant criteria for closed holomorphic curves to be disjoint and/or embedded. The subject of this minicourse is the generalization of these tools to punctured holomorphic curves with cylindrical ends in 4-dimensional symplectic cobordisms. First developed in Siefring's thesis around 2005 (and building on earlier work of Hutchings), this theory has become a powerful tool for proving rigidity results in 3-dimensional contact topology and dynamics. The main difficulty compared with the closed case is that the intersection number of two curves (which is not even obviously finite a priori) need not be homotopy invariant, but a precise understanding of the asymptotic behavior of such curves shows that a homotopy-invariant intersection product can be constructed by including a count of "hidden intersections at infinity". We will try to present the essential ideas of this theory in a maximally user-friendly form, and then demonstrate its use in a few applications, including classification of symplectic fillings, and defining a version of contact homology for the complement of a set of Reeb orbits.

(scanned notes provided by Alex Cioba)

Notes:

 Main references:

Orderability and Rabinowitz Floer theory (Peter Albers and Will Merry)

Overview:
We present a link between the notion of orderability introduced by Eliashberg-Polterovich in 2000 and Hamiltonian perturbations of Rabinowitz Floer homology (RFH). After explaining the definition and basic properties of RFH we first establish a connection to leafwise intersections and, in particular, translated points. Following Sandon's approach to contact rigidity we use RFH to detect orderability. Moreover, we point out a relation to the Weinstein conjecture. In the last lecture we continue by defining a contact capacity (in the sense of Sandon) and prove an abstract non-squeezing result. As an application we reprove the beautiful non-squeezing results of Eliashberg-Kim-Polterovich. Main references:

Titles and abstracts for the research talks

schedule     minicourse synopses     titles/abstracts for research talks

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