Fukaya categories of complex symplectic manifolds

这个项目关注的是一类叫做辛分辨率的数学对象。辛分辨率是数学上的瑰宝；它们拥有大量有趣的结构，这些结构引起了不同领域的数学家和理论物理学家的兴趣。因此，关于辛分辨率的新发现往往对数学和物理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（连同合作者）将研究与称为深谷范畴的辛分辨率相关的某些代数结构。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将编写面向高级本科生或初级研究生的公开课堂讲稿。辛流形的Fukaya范畴是一类通过非线性分析，通过计算具有适当拉格朗日边界条件的伪全纯曲线而定义的代数结构。长期以来，人们一直期望深谷的辛决议类别应该与表征理论中产生的结构有关。PI打算利用微局部轴理论的技术，在这个方向上研究几个问题。多亏了Ganatra- Pardon- Shende的基础工作，这些技术最近才变得可行，PI相信它们对这类问题特别有希望。在此过程中，PI还计划进一步发展微局部轴理论的基础，特别是反常微局部轴理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.