Differential graded categories and their applications in geometry

微分分级范畴及其在几何中的应用

• 批准号: 0707210
• 负责人: Dmitry Tamarkin
• 金额: $ 11.65万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707210&HistoricalAwards=false
• 关键词: Differential; graded; categories; their; applications

The goal of this project is to study and develop applications of the apparatus of differential graded categories (DG) to several problems of geometry and mathematical physics. First, the PI would like to develop a DG version of the microlocal theory of constructible sheafs due to Kashiwara-Schapira and then, using this apparatus, given a symplectic manifold, he would like to construct a dg-category and to compare it with the Fukaya category.The PI would also like to continue his work on mathematically rigorous Poisson sigma-model, non-commutative calculus (with his collaborators), and theory of n-categories.Differential graded (DG) categories play an important role in a variety of problems in algebraic geometry and mathematical physics. The goal of this project is to further develop some applications of DG categories. One of these applications is aimed at providing an alternative construction of the celebrated Fukaya category, this construction is supposed to be more geometric and transparent in spirit. The PI would also like to continue on several topics from his previous research such as mathematically rigorous models of quantum field theory, non-commutative calculus (with his collaborators), and further development of the apparatus of DG categories.

本计画的目标是研究及发展微分分次范畴（DG）装置在几何与数学物理中的应用。 首先，PI想发展由Kashiwara Schapira提出的可构造层的微局部理论的DG版本，然后，使用这个装置，给定一个辛流形，他想构造一个DG-范畴，并将其与福谷范畴进行比较。PI还想继续他在数学上严格的Poisson sigma模型，非交换微积分，微分分次范畴（DG）在代数几何和数学物理中的许多问题中起着重要的作用。本项目的目标是进一步开发DG类别的一些应用。这些应用之一是旨在提供一个著名的福谷类别的替代结构，这种结构应该是更几何和透明的精神。PI还想继续他以前研究的几个主题，如量子场论的数学严格模型，非交换微积分（与他的合作者），以及DG类别设备的进一步发展。