Tag - Contact geometry

In this talk we consider the links of simple singularities, which are contactomoprhic to S³/G for finite subgroups G of SU₂(ℂ). We explain how to compute the cylindrical contact homology of S³/G by means of perturbing the canonical contact form by a Morse function that is invariant under the corresponding rotation subgroup. We prove that the ranks are given in terms of the number of conjugacy classes of G, demonstrating a form of the McKay correspondence. We also explain how our computation realizes the Seifert fiber structure of these links.

We compute the homotopy type of any connected component of the space of tight contact structures on a 3-fold. In fact, we actually prove a partial h-principle for the inclusion of the contactomorphism group into the diffeomorphism group. The basic building block is the homotopy equivalence induced by the inclusion of the contactomorphism group of the sphere relative to a point and the diffeomorphism group relative to a point result recently proven by Elisahberg and Mishachev.

Then, we wonder how these sets of techniques work for overtwisted manifolds? i.e. we just try to prove the same theorem than in the tight case, assuming that the triangulation is very small, we easily obtain that all the cells are tight and then, everything looks like working, so however, there must be something wrong because we find several contradictions: the overtwisted mirage. Once the mistake is understood, we proceed to compute the homotopy type of the space of contact structures/contactomorphisms by using just Mishachev-Eliiashberg result, i.e. we reprove the 3-dimensional overtiwsted h-principle as a corollary. We will compute the space of embeddings of overtwisted disks in some particular manifolds. Finally we end by explaining the conjecture tight overtwisted.

In this talk, we will discuss the Chiu-Tamarkin complex. It is a symplectic/contact invariant that comes from the microlocal sheaf theory. I will explain how to define some capacities using the Chiu-Tamarkin complex in both symplectic and contact situations. The main result is the structure theorem of the Chiu-Tamarkin complex of convex toric domains. Consequently, we can compute the capacities of convex toric domains.

In this talk we will discuss invariants of sutured Legendrians. A sutured contact manifold can be seen as either generalizing the contactisation of a Liouville domain, or as a presentation of a contact manifold with convex boundary. Using the first point of view, we define the wrapped sutured homology of Legendrians with boundary, employing ideas coming from Floer theory. To illustrate the second aspect, we apply the unit conormal construction to braids with two strands, which yields a sutured Legendrian. We will show that, if the conormals of two 2-braids are Legendrian isotopic, then the braids are equivalent.

In this talk, we will discuss the Chiu-Tamarkin complex. It is a symplectic/contact invariant that comes from the microlocal sheaf theory. I will explain how to define some capacities using the Chiu-Tamarkin complex in both symplectic and contact situations. The main result is the structure theorem of the Chiu-Tamarkin complex of convex toric domains. Consequently, we can compute the capacities of convex toric domains.

Inspired by the symplectic Banach-Mazur distance, proposed by Ostrover and Polterovich in the setting of non-degenerate starshaped domains of Liouville manifolds, we define a distance on the space of contact forms supporting a given contact structure on a closed contact manifold and we use it to bi-Lipschitz embed part of the 2-dimensional Euclidean space into the space of overtwisted contact forms supporting a given contact structure on a smooth closed manifold.

Is every dynamically convex contact form on the three sphere convex? In this talk I will explain why the answer to this question is no. The strategy is to derive a lower bound on the Ruelle invariant of convex contact forms and construct dynamically convex contact forms violating this lower bound.

I will explain the construction of a functor from the exact symplectic cobordism category to a totally ordered set, which measures the complexity of the contact structure. Those invariants are derived from a bi-Lie infinity formalism of the rational SFT and a partial construction of the rational SFT. In this talk, I will focus on the construction and properties of the functor. Time permitting, I will explain applications, computations, and relations to the involutive bi-Lie infinity formalism of the full SFT. This is joint work with Agustin Moreno.