Fukaya categories of complex symplectic manifolds

这个项目涉及一类被称为辛分解的数学对象。辛分解是数学上的宝石；它们拥有丰富有趣的结构，这是从事不同领域工作的数学家和理论物理学家感兴趣的。因此，关于辛分辨率的新发现往往对数学和物理学的许多部分产生广泛的影响。作为一名在辛几何领域工作的数学家，PI(与合作者一起)将研究与辛分解相关的某些代数结构，称为Fukaya范畴。PI还将指导希望进入这一数学领域的本科生和研究生。为此，PI将为高级本科生或初级研究生撰写公开可用的课堂讲稿。Fukaya辛流形是通过非线性分析定义的一类代数结构，通过计算具有适当拉格朗日边界条件的伪全纯曲线。长期以来，人们一直期望Fukaya辛分解范畴应该与表示理论中出现的结构相关。PI打算利用微局部鞘理论的技术，在这个方向上研究几个问题。这种技术直到最近才出现，这要归功于加纳特拉--原谅--申德的基础工作，国际刑警组织认为，对于这类问题，它们特别有希望。在此过程中，PI还计划进一步发展微局部鞘理论的基础，特别是倒置微局部鞘理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with a class of mathematical objects called symplectic resolutions. Symplectic resolutions are mathematical jewels; they possess a wealth of interesting structures, which are of interest to mathematicians and theoretical physicists working in different areas. As a result, new discoveries about symplectic resolutions often have a wide impact across many parts of mathematics and physics. As a mathematician working in an area called symplectic geometry, the PI (along with collaborators) will study certain algebraic structures associated to symplectic resolutions called Fukaya categories. The PI also will mentor undergraduate and graduate students wishing to enter this area of mathematics. To this end, the PI will write publicly available lecture notes aimed at advanced undergraduate or beginning graduate students.Fukaya categories of symplectic manifolds are a class of algebraic structures defined via nonlinear analysis, by counting pseudoholomorphic curves with appropriate Lagrangian boundary conditions. There are long-standing expectations that Fukaya categories of symplectic resolutions should be related to structures arising in representation theory. The PI intends to work on several questions in this direction, using techniques from microlocal sheaf theory. Such techniques have only recently become available thanks to the fundamental work of Ganatra--Pardon--Shende, and the PI believes that they are particularly promising for these types of questions. Along the way, the PI also plans to further develop the foundations of microlocal sheaf theory, in particular the theory of perverse microlocal sheaves.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.