Tag - Contact geometry

Peter Albers: Positive loops-on a question by Eliashberg-Polterovich and a contact systolic inequality

25th February, 2016

In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether C0-small positive loops exist. We give a negative answer to this question. Moreover we give sharp lower bounds for the size which, in turn, gives rise to a L∞-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no L2-contact systolic inequality exists. The choice of L2 is motivated by systolic inequalities in Riemannian geometry.

Kai Zehmisch: Disc filling and connected sum

27th February, 2015

In my talk I will report on recent work with Hansjörg Geiges about a strong connection between the topology of a contact manifold and the existence of contractible periodic Reeb orbits. Namely, if the contact manifold appears as non-trivial contact connected sum and has non-trivial fundamental group or torsion-free homology, then the existence is ensured. This generalizes a result of Helmut Hofer in dimension three.

Stefan Müller: C0-characterization of symplectic and contact embeddings

7th November, 2014

Symplectic and anti-symplectic embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C0-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C. Sikorav and Y. Eliashberg). The aforementioned rigidity is again an easy consequence. The shape invariant has two immediate advantages: it avoids the cumbersome distinction between symplectic and anti-symplectic, and the results can be adapted to contact embeddings via coisotropic embeddings (of tori). The adaptation to the contact case is (partly) work in progress. In the talk, I will give proofs of the main results and explain what makes them work (J-holomorphic disc techniques for all deep results). I plan to also touch upon a theorem of Sikorav that characterizes subsets of the unit cotangent bundle of a torus in terms of the shape invariant (plus possible generalizations). As much as time permits, I will end the talk with a brief survey of some of my related results.

Mark McLean: Minimal Discrepancy of Isolated Singularities and Reeb Orbits

4th April, 2014

Let A be an affine variety inside a complex N-dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization.

Keon Choi: New combinatorial computations of embedded contact homology

7th March, 2014

Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and the almost complex structure on its symplectization. This fact can be used to extract symplectic geometric information (e.g. ECH capacities) but explicit computation of the chain complexes has been carried out only on a few cases. Extending the work of Hutchings-Sullivan, we combinatorially describe the ECH chain complexes of T3 with general T2-invariant contact forms and certain almost complex structures.

Michael Hutchings: How not to define cylindrical contact homology

18th October, 2013

We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. One can fix this difficulty by using S1-dependent almost complex structures, but at the expense of introducing another difficulty which we will explain. We outline how fixing the latter difficulty ultimately leads to a different theory, an analogue of positive symplectic homology. This talk is intended to be part of a series of expository talks on the foundations of contact homology, but prerequisites should be minimal.

Peter Weigel: Positive loops and orderability in contact geometry

4th October, 2013

Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional Σ so that the result is always orderable. The main technical tool is a Morse-Bott Floer-theoretic growth rate, which has some parallels with Givental's non-linear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(Σ).