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First of all, the Vianna tori are distinguished by the Fukaya category. To get an object of the category, you need to pick a torus and a rank 1 local system on it. The set of local systems making the …

One goes hunting for Lagrangian submanifolds if one is interested in defining gauge theoretic 3-manifold invariants by cutting along a Heegaard surface. Lagrangian submanifolds arise in gauge theory …

Another place where Lagrangian submanifolds arise naturally is in cotangent bundles. If $M\subset N$ is a submanifold then you can define the conormal bundle of $M$ to be the set of covectors which an …

Sometimes it is, sometimes it isn't. The spheres living over the edges generate the second homology, so you can read off the action on $H_2$ from that. For $S^2\times S^2$ (square) the action on homol …

One classic thing to do is to take a sequence of holomorphic curves with a tangency condition and assume that the tangency condition still holds in the limit: if the limit is a multiple cover with a b …

There is a spectral sequence from the cohomology of the intersection to the Floer cohomology. This is explained (and used) in Seidel's paper "Lagrangian 2-spheres can be symplectically knotted" https …

By the time I finished writing this answer someone has explained the whole idea in a comment, but I thought I'd post it anyway as there is some more detail here. I assume the version of Arnold-Liouvil …

Neck-stretching is a deformation of an almost complex structure in a neighbourhood of a hypersurface. In the Eliashberg-Givental-Hofer paper, neck-stretching is called "splitting along a contact subma …

Here's a complement to Dmitri's excellent answer. While it's true that contactomorphisms can have fixed points, even for some open set of contactomorphisms, there is a reason you don't expect to be ab …

This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (am …

Just to explicitly answer the first part of your question, the original version of Nemirovski's first paper (https://arxiv.org/abs/math/0106122v1) surveys what is known about the other surfaces. Namel …

When you try and prove that $d^2=0$ ($d$ being the Floer differential) you need to look at the boundary of the moduli space of index 2 J-holomorphic strips with one boundary on $L_0$, one on $L_1$. Ce …

In this case, the J-holomorphic curves are all constant, so the evaluation pseudocycle is the tridiagonal $\{(x,x,x) : x\in M\}$. You take cycles $A_1,A_2,A_3$ Poincare dual to $a_1,a_2,a_3$ respectiv …