Flexible Stein Manifolds and Fukaya Categories

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The PI is studying a class of geometric objects called symplectic manifolds, which are related to classical mechanics, modern physics and string theory, as well are pure mathematics and topology. In the past decades there has been a large trend toward studying symplectic manifolds using a tool called Fukaya categories, which measures the geometry and packages the information into algebraic data which can be quantified and understood more readily than the geometry itself. This project aims to study the extent to which the Fukaya category is complete as a measurement, or loses information. The research will focus on a particularly well-understood case, that of flexible manifolds. The tools the PI proposes for studying this question come from a number of recent discoveries in the field. Results from this project will be particularly useful for bridging the gaps in our our understanding between the geometry of symplectic manifolds, and their Fukaya categories.Some of the tools the PI intends to use come from constructable sheaf theory, partially wrapped Fukaya categories, arboreal singularities, and loose Legendrians. A naive conjecture is that, if the wrapped Fukaya category of a Weinstein manifold is trivial, then the manifold is flexible. Though this is known to be false (due to the existence of subflexible manifolds), related statements are known to be true by strengthening the hypotheses. A major bridge the PI proposes is the use of arboreal singularities, where flexibility is comparatively easy to detect. Additionally arboreal singularities have the advantage that the Fukaya category can be expressed with finite combinatorial methods, here using isomorphisms between Floer homology and microlocal sheaf theory. Various special cases, such as the case of contractible 6-manifolds, would yield strong results and are potentially easier to approach.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.