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Hamiltonian systems, symplectic flows, classical integrable systems
2
votes
Why we have to fix markers in SFT?
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 α be a contact 1-form …
3
votes
SFT compactness
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 …
14
votes
Accepted
Kuranishi structures vs polyfolds
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 …
11
votes
Accepted
Orientations for pseudoholomorphic curves with totally real boundary condition
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 It was solved Vin de Silva in his (unpublished) D. Phil …
3
votes
Accepted
Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds
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 Sn …
6
votes
Accepted
A Poisson Geometry Version of the Fukaya Category
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 …
7
votes
Hamiltonian circle actions and Lefschetz pencils
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 …
6
votes
Accepted
The higher structure of the Floer cochains of the diagonal in CP^ x CP^n
Here's an argument that the diagonal Lagrangian correspondence Δ in CPn×CPn is formal. That is, its Floer cochains CF∗(Δ,Δ), as an A∞-algebr …
24
votes
Accepted
What is the Poincare dual of a symplectic form?
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 …
6
votes
Are there symplectic 4-folds with b+>1, b−=0?
Symplectic geography in 4 dimensions can be mapped using Chern number coordinates (c21,c2). The part of the plane where c21>4c2 is uncharted. It's unknown whether there are any symplectic …
11
votes
Accepted
symplectic 4-manifolds with free circle action
Here's an example, using a construction of Fernandez, Gray and Morgan (1991):
Take a closed surface S with area form ω, let ϕ be an area-preserving diffeomorphism, and $p\colon S_\phi …
11
votes
Real interpretations of Discontinuities in Floer homology
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=S1×y as a Lagrangian in standard symplectic $T^2=S^1\time …
20
votes
When are two symplectic forms "isotopic"?
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 …
8
votes
Accepted
Length of Floer flow lines
In your symplectically aspherical setting, bounds on length will indeed exist.
Suppose one has a sequence of solutions un to Floer's equation, of bounded energy, and a sequence of points $t_n\in …
14
votes
Accepted
Obstruction bundle for spaces with Kuranishi structure
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 …