Has anything precise been written about the Fukaya category and Lagrangian skeletons? At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons."  As I understand it, a
Lagrangian skeleton is a union of Lagrangian submanifolds which a
symplectic manifold retracts to.  One good example would be the
zero-section of a cotangent bundle, but there are others; for example,
the exceptional fiber of the crepant resolution of $\mathbb
C^2/\Gamma$ for $\Gamma$ a finite subgroup of $SL(2,\mathbb C)$.  From the rumors I've heard, apparently there's some connection between the geometry of the skeleton and the Fukaya category of the symplectic manifold; this is understood well in the case of a cotangent bundle from work of Nadler and Nadler-Zaslow
I'm very interested in the Fukaya categories of some manifolds like
this, but the only thing I've actually seen written on the subject is
Paul Seidel's moderately famous picture of Kontsevich carpet-bombing
his research program

which may be amusing, but isn't very
mathematically rigorous.  Google searching hasn't turned up much, so I
was wondering if any of you have anything to suggest.
 A: On this topic I've only seen Ganatra's notes from Paul's Talbot talk (see Scott's answer). An aspect of this, that a Fukaya category can behave in a sheafy way, is part of Nadler's proof that "microlocal branes are constructible sheaves".
The most relevant reference I can think of for the Fukaya category of the crepant resolution of $\mathbb{C}^2/\Gamma$ is Abouzaid's paper about Floer theory in plumbings.
Bad form as it may be, I'll ask a reciprocal question in this answer box. Suppose you knew that the Fukaya category of this crepant resolution was the constructible derived category on the exceptional fibre (was that what you had in mind?). What could one deduce? Would there then be a quiver presentation? A reasonable description of Hochschild cohomology?
A: I asked Paul a couple months ago, and he said that nothing has been written, but there may be notes floating around from some talks Kontsevich gave in France last March.  There are a few examples given in the last talk at Talbot on the bottom of this page.  I was unable to work out a precise mathematical statement from the information given, but I'm not a symplectic geometer.
A: I noticed this question has been bumped up to the front page, and the
most recent answer is about 8 years old: the subject has moved on
since then, and more has been written. Here is my understanding of
some of the recent developments.
Caveat: This whole area has undergone rapid progress in the last few
years, and I am not working on this question, so what I say is
probably not up-to-date (even as I write, never mind in the future).
Kontsevich's idea was that you can relate the Fukaya category of a
Weinstein manifold to the microlocal sheaf category of its
skeleton. As far as I know, a proof of this conjecture is work in
progress by Ganatra-Pardon-Shende (one preliminary part of which is
already available).
I think the idea of their proof is roughly the following:

*

*You prove that the Fukaya category has a co-sheaf property, which
means it can be computed locally first on some subsets and then the
answer can be glued together using homotopy colimits. For this, you
need suitable functors relating the Fukaya category of a subset to
the Fukaya category of the whole manifold. This is complicated by
the fact that your "subsets" might not be very nicely embedded in the
whole manifold: for example, if your ambient manifold is $T^*M$ then
you want to allow subsets like $T^*M'$ where $M'\subset M$ is a
codimension zero submanifold with boundary (that's because you
ultimately want to work locally on the skeleton). That is
problematic because the Liouville vector field for $T^*M'$ and the
Liouville vector field for $TM$ don't match up nicely. The first GPS
paper constructs these categories and functors for "Liouville
sectors" (a suitably broad class of inclusions, related to Sylvan's
notion of stops and partially wrapped Floer homology). I think the proof of the co-sheaf property is still ongoing work?


*Now you compute the local pieces of the Fukaya category and show
that they agree with the microlocal sheaf categories; since both
have co-sheaf gluing, you get the same global answers.
The second part relies on some local computations of Fukaya
categories. Nadler has introduced the
notion of "arboreal skeleton" which is a skeleton with certain
"generic" singularities. For example, trivalent graphs in dimension 1;
trivalent graph times interval or cone on 1-skeleton of a tetrahedron
in dimension 2; etc. He computes the microlocal sheaf category for
these; I'm not sure if the corresponding partially-wrapped Fukaya
categories have been calculated in all cases yet. Finally, you want to
show that any Weinstein manifold has an arboreal skeleton:
Starkston has some results in this
direction which may represent the state of the art.
Leaving this aside for a moment, there are also special cases where
the Konstevich conjecture/local-to-global results for the Fukaya
category has been established independently of this general
program. These include (but again, I'm probably missing some):

*

*Nadler https://arxiv.org/abs/1601.02977

*Pascaleff-Sibilla https://arxiv.org/abs/1604.06448

*Nadler https://arxiv.org/abs/1604.00114

*Lee https://arxiv.org/abs/1608.04473

*Gammage-Nadler https://arxiv.org/abs/1702.03255

*Shende https://arxiv.org/abs/1707.07663
A: A comment re. Jonny's nice answer: there was indeed a time when that was the envisioned strategy of proof.  However our present approach does not require the arborealization.  Because: now we know that the Fukaya and microlocal categories associated to any (singular) Legendrian in a cosphere bundle agree.
--
In GPS2, we prove descent for general sectorial covers.  In particular, one could deduce the existence of said cosheaf by an (unwritten) purely geometric argument that an open cover of the skeleton lifts to a sectorial cover of the symplectic manifold.  However, the notion of sectorial cover is more flexible, as it is not tied to a particular skeleton; in a certain sense it captures all skeleta at once.
The real virtue of the cosheaf on the skeleton is that it is not some arbitrary cosheaf of categories, but in fact is the Kashiwara-Schapira stack of microlocal sheaf theory. (Historical note: this assertion goes beyond the original conjecture of Kontsevich, and is probably best attributed to Nadler.) As mentioned above, GPS3 is the required local
calculation.  In principle, one could use GPS2 to glue, but actually there is a short-cut for most Weinstein manifolds of interest in mirror symmetry and geometric representation theory.  The reason is that these manifolds typically have trivial stable symplectic normal bundle, so one can use the embedding trick without messing about with descent from a Lagrangian Grassmannian bundle. This reduces the problem to the cotangent comparison of GPS3, provided one has certain full faithfulness of embedding results in both partially wrapped Floer theory and in microlocal sheaf theory.
Full faithful embeddings come from a certain doubling construction, which was originally introduced in sheaf theory by Guillermou, it is described in e.g. section 11.4 of his omnibus.  Guillermou assumes there that the relevant skeleton is smooth, but similar ideas give a construction in general and appear in Sec. 6 of my paper with Nadler.  On the wrapped Floer theory side the construction is Example 8.6 in GPS2.  Combining the above one learns that for a Weinstein manifold with stably trivial symplectic normal bundle, the wrapped Fukaya category is equivalent to the category of microlocal sheaves on any Lagrangian skeleton.
(Most likely the `embedding trick' also works to establish the result in the general case, but this requires a bit more complicated argument.)
A: There are a few comments about this in the notes from Kontsevich's talk (Internet Archive) at the Arbeitstagung (Internet Archive).
I also attended the Seidel talk that people are referring to. I should have my notes somewhere; if I succeed in finding them and if they contain anything not already in Sheel Ganatra's notes, I'll scan them and post them.
Edit: Kontsevich spoke about this last week in Miami. Check out these notes from the talk. Find the abstract here.
A: Not sure if this is helpful to you, but Bing came upon this paper by Ruan which I'm thinking you likely have already seen and "get."
The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold
