Flexible Stein Manifolds and Fukaya Categories

PI正在研究一类称为辛流形的几何对象，这与经典力学，现代物理学和弦理论有关，也是纯数学和拓扑学。在过去的几十年里，有一个很大的趋势是使用一种叫做福谷范畴的工具来研究辛流形，这种工具可以测量几何形状并将信息打包成代数数据，这些数据比几何形状本身更容易量化和理解。这个项目的目的是研究福谷类别作为一个测量是完整的，或丢失信息的程度。研究将集中在一个特别好理解的情况下，即柔性流形。PI提出的研究这个问题的工具来自该领域的一些最新发现。从这个项目的结果将是特别有用的桥梁之间的差距，在我们的理解辛流形的几何形状，和他们的福谷categories。一些工具的PI打算使用来自可构造层理论，部分包裹福谷类别，树奇异性，和松散的Legendrians。一个朴素的猜想是，如果一个Weinstein流形的包裹福谷范畴是平凡的，那么这个流形是柔性的。虽然这被认为是错误的（由于亚柔性流形的存在），但通过加强假设，相关的陈述被认为是正确的。PI提出的一个主要桥梁是使用树奇异点，其中灵活性相对容易检测。另外树奇异性的优点是福谷范畴可以用有限的组合方法表示，这里使用弗洛尔同调和微局部层理论之间的同构。各种特殊情况下，如可收缩的6-流形的情况下，将产生强有力的结果，并可能更容易接近。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.