Fukaya categories of complex symplectic manifolds

复辛流形的深谷范畴

• 批准号: 2305257
• 负责人: Benjamin Gammage
• 金额: $ 19.49万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305257&HistoricalAwards=false
• 关键词: Fukaya; categories; complex; symplectic; manifolds

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.

该项目与称为符号分辨率的一类数学对象有关。符号分辨率是数学珠宝；他们拥有大量有趣的结构，这对于在不同领域工作的数学家和理论物理学家感兴趣。结果，关于符号决议的新发现通常在数学和物理学的许多部分都产生广泛的影响。作为一名在名为Symbletectic几何的领域工作的数学家，PI（与合作者一起）将研究与称为福卡亚类别的符合性分辨率相关的代数结构。 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将撰写针对高级本科或初学者的公开讲义。Fukaya类别类别是通过非线性分析来定义的一类代数结构，通过计算具有适当Lagrangian Lagrangian边界条件的伪酚晶状体曲线。长期以来的期望，即福卡亚类别的符号分辨率类别应与代表理论中产生的结构有关。 PI打算使用Microleocal Sheaf理论的技术在这个方向上处理几个问题。由于Ganatra-Pardon-Shende的基本工作，此类技术直到最近才获得，并且PI认为它们对于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束缚理论的基础，尤其是不正当的微局部滑轮理论。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估标准来通过评估来支持的。