The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology.
McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.
I'll outline recent results with Steven Sivek classifying the Stein fillings, up to topological homotopy equivalence, of the canonical contact structure on the unit cotangent bundle of a surface. The proof begins with Li, Mak and Yasui's technology for Calabi-Yau caps.
We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone S2×S2. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.
We present some homological mirror symmetry statements for the singularities of type Tp,q,r. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E. We will consider some symplectic invariants of the real 4-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space P2, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different 'flavours' of invariants (e.g., versions of the Fukaya category) match up on both sides.
After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called 'subflexible', which are not flexible but become so after attaching some Weinstein handles. For example, the standard symplectic ball has a Weinstein subdomain with non-trivial symplectic topology. These are exotic symplectic manifolds with vanishing symplectic cohomology. I will explain how to study them using a deformed version of symplectic cohomology, and how this invariant can computed using the machinery of Fukaya categories and Lefschetz fibrations. This is partly based on joint work with Emmy Murphy.
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.
I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I will explain the consequences of quantum cohomology, and the expected applications of this theory.
In this talk we first introduce a new 'singularity-free' approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including ℝPn and ℂPn, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a 'mirror' of the construction in algebraic geometry, realized by a new surgery and cobordism construction.
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.
You must be logged in to post a comment.