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The first case has finite indices and parabolic gradient flow; the second infinite (co)indices and elliptic gradient flow. In more detail, the Morse theory of the energy functional $E$ on $X:=\Omega( …

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 It was solved Vin de Silva in his (unpublished) D. Phil …

In his thesis http://arxiv.org/abs/math/0401221 Marco Gualtieri explains that a generalized almost complex structure on an $n$-manifold $M$ is a reduction of the structure group of $TM \oplus T^\ast …

When $(X,\omega)$ is a near-symplectic oriented 4-manifold there is always a canonical identification between $\mathrm{Spin}^c(X)$ and the classes in $H_2(X,Z;\mathbb{Z})$ that bound $[Z]$, where $Z=\ …

If your CY manifold is simply connected, the base of the torus-fibration will have to be simply connected too, since a homotopically non-trivial loop downstairs would lift to a loop upstairs which doe …

There is a tacit assumption behind this question, which I don't think is justified: that the Abel-Jacobi images of the Heegaard tori $\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$ are Lagrangian with …

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 …

I'll comment on the related question "what is the Serre functor for the Fukaya category?" Calabi-Yau setting The Serre functor $S$, by definition, satisfies $\mathsf{Hom}(X,SY) \cong \mathsf{Hom}(Y …

I know of no technique capable of bounding above the homotopy groups of a symplectomorphism group in dimension $\geq 6$, nor of a contactomorphism group in dimension $\geq 5$. There are, however, te …

A biased answer, based on Auroux's work http://arxiv.org/abs/1003.2962. Auroux makes a connection between bordered Floer theory and an alternative approach, due to Lekili and myself, which is (still …

Some blah on symplectic homology vs. cohomology. There's an invariant $SH(M)$ of Liouville domains $M$ which some people call symplectic homology and some symplectic cohomology. This is the direct lim …

I wrote a little expository piece about this and related matters in the Newsletter of the European Mathematical Society: http://www.ems-ph.org/journals/newsletter/pdf/2010-03-75.pdf The classical to …

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 …

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 …

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 …