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I'll talk about cylindrical contact homology, which is a relatively well-established part of SFT. It's a variant of Hamiltonian Floer homology for contact manifolds. Let $\alpha$ be a contact 1-form …

For non-specialist readers: SFT = symplectic field theory BEHWZ = Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder, the authors of the paper which establishes the basic compactness theorem for pseudo-hol …

Kuranishi models are a traditional - and beautiful - technique for describing the local structure of moduli spaces cut out by non-linear equations whose linearization is Fredholm. A more elaborate ver …

1) The problem of orienting moduli spaces of pseudo-holomorphic discs with totally real boundary conditions is really a problem in index theory. It was solved Vin de Silva in his (unpublished) D. Phil …

jc, you'll have fun working out answers to examples of your first question. I'm only going to address the Legendrian case. If I didn't make mistakes, I'll conclude that Legendrian immersions of $S^n$ …

The fundamental technique of symplectic topology is the theory of pseudo-holomorphic curves. One studies maps $u$ from a Riemann surface into a symplectic manifold, equipped with an almost complex str …

Hm, good question. This would be potentially interesting from the point of view of classifying Hamiltonian circle-actions, "by induction" on the dimension. (1) I'm pretty sure that the answer is not …

Here's an argument that the diagonal Lagrangian correspondence $\Delta$ in $\mathbb{C}P^n \times \mathbb{C}P^n$ is formal. That is, its Floer cochains $CF^\ast(\Delta,\Delta)$, as an $A_\infty$-algebr …

One of the big advances in symplectic topology in the 90s was Donaldson's theorem that when the symplectic class is integral, high multiples of its dual are represented by symplectic submanifolds. T …

Symplectic geography in 4 dimensions can be mapped using Chern number coordinates $(c_1^2,c_2)$. The part of the plane where $c_1^2 > 4c_2$ is uncharted. It's unknown whether there are any symplectic …

Here's an example, using a construction of Fernandez, Gray and Morgan (1991): Take a closed surface $S$ with area form $\omega$, let $\phi$ be an area-preserving diffeomorphism, and $p\colon S_\phi …

The brief answer is yes, using ideas from Novikov homology. Here's an example of the discontinuity and how it can be fixed. Take $L=S^1\times y$ as a Lagrangian in standard symplectic $T^2=S^1\time …

There is a cheap way to find cohomologous but non-isotopic (in fact, non-deformation equivalent) symplectic forms: start with a symplectic manifold and pull back the symplectic form via a diffeomorphi …

In your symplectically aspherical setting, bounds on length will indeed exist. Suppose one has a sequence of solutions $u_n$ to Floer's equation, of bounded energy, and a sequence of points $t_n\in …

Here's a view of the symplectic side of the bridge. The Kuranishi model (see Donaldson-Kronheimer, The geometry of four-manifolds, ch. 4) goes like this. You're interested in a (moduli) space $M$ cut …