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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 …

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 …

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomo …

V. I. Arnol'd, June 12, 1937 - June 3, 2010. The very sad news of his death is reported today here. After Floer, the main difficulty in solving the weak Arnol'd conjecture on a compact symplectic ma …

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 …

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( …

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 …

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 …

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 can't speak for these authors, but what I understand by a "Fourier-Mukai" transform between Fukaya categories is the functor between extended Fukaya categories associated with a Lagrangian correspon …

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 …

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 It was solved Vin de Silva in his (unpublished) D. Phil …

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 …

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 …