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I am interested in Lagrangian correspondences in the context of symplectic manifolds, namely Lagrangian submanifolds $L_{12}$ of $M_1\times \bar M_2$ where $M_1$ and $M_2$ are symplectic manifolds with symplectic forms $\omega_1$ and $\omega_2$, and $\bar M_2$ has its symplectic form reversed (so the symplectic form on the product is written $\omega_1-\omega_2$, if we omit tedious pullbacks).

As experts know better than I, these correspondences are to be thought of as morphisms in a putative symplectic category. Unfortunately, producing a correspondence $L_{13}$ given correspondences $L_{12}$ and $L_{23}$ is not obvious, unless the latter two intersect transversally.

As of ~10 years ago, there were a few candidate approaches to resolve the issue, which were summarised by Weinstein here:

Weinstein, Alan, Symplectic categories, Port. Math. (N.S.) 67, No. 2, 261-278 (2010). ZBL1193.53173.

What is the state of the art today? Has there been any progress in finding a useful and versatile definition that does not suffer from (as Weinstein calls it) this nontransversality problem?

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Though not about solving the nontransversality problem, Fukaya's paper Unobstructed immersed Lagrangian correspondence and filtered A infinity functor is the state of the art in why nontransversality isn't a problem for most interesting purposes. It's been known for a long time that generic compositions are immersed, so Fukaya just works with immersed Lagrangians equipped with bounding cochains and shows that this data can be carried through a correspondence to give an $A_\infty$-functor on Fukaya categories.

You should also check out the work of Nate Bottman, much of which is motivated by trying to construct higher categorical versions of Weinstein's category. The survey article by Abouzaid and Bottman should give you a very up to date picture of the field: Functoriality in categorical symplectic geometry.

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  • $\begingroup$ Thanks Jonny for both references. The "most interesting purposes" you mention, these are presumably the construction of the A-infinity category in Fukaya? My own interest comes from physics/string theory (although not from type A topological strings originally; the 2nd paper you linked has some suggestive images though) where the nontransversality seems to lead to fatal issues with the dynamics, which I would conjecture are analogous to issues involving "colliding seams" in the context of the pseudoholomorphic quilts described in Abouzaid and Bottman $\endgroup$
    – AlexArvanitakis
    Commented Dec 20, 2022 at 23:14
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    $\begingroup$ The interesting applications I was referring to come from the construction of A_\infty functors between Fukaya categories, not the categories themselves (which do not require correspondences to set up). Colliding seams happen in Fukaya's paper, and the main idea he uses to deal with them is the "Y-end" of Lekili-Lipyanskiy: arxiv.org/abs/1003.4493 $\endgroup$
    – Jonny Evans
    Commented Dec 21, 2022 at 0:30
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    $\begingroup$ (Colliding seams occur whenever you try and understand Floer theory under composition of Lagrangian correspondences, whether the composition is transverse or not) $\endgroup$
    – Jonny Evans
    Commented Dec 21, 2022 at 0:33

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