Properties and Applications of the Microlocal Category

微局部类别的属性和应用

• 批准号: 1612437
• 负责人: Dmitry Tamarkin
• 金额: $ 16.52万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612437&HistoricalAwards=false
• 关键词: Properties; Applications; Microlocal; Category

The geometric structure that lies behind classical Newtonian mechanics is called symplectic geometry, where the coordinates of position and momentum, along with the conservation laws of mechanics, are modeled in even-dimensional geometries with distinctive structures. While the familiar geometry of Euclidean spaces takes length and angle as fundamental notions, the structure of symplectic geometry begins from the measurement of two-dimensional area. Much of the modern study of symplectic geometry is devoted to understanding distinguished structures on symplectic manifolds, a study that can reveal useful invariants for those manifolds. The research to be pursued in this project is focused on questions in this area arising out of string theory.This research project focuses on developing the theory of microlocal categories, such as the full sub-category of objects supported on a Lagrangian submanifold, and on developing an appropriate ground category for the microlocal category, which may be topological in origin. The project also aims to find a description of the microlocal categories associated to flag varieties. Further investigations are motivated by ideas of quantization, which is here understood in geometric terms abstracted from concepts in physics.

经典牛顿力学背后的几何结构被称为辛几何，其中位置和动量的坐标以及力学守恒定律被建模为具有独特结构的偶数维几何。常见的欧几里得空间几何以长度和角度为基本概念，而辛几何的结构始于二维面积的测量。现代辛几何的许多研究都致力于了解辛流形上的区别结构，这一研究可以揭示这些流形上有用的不变量。本课题的研究重点是弦理论在这一领域的研究，重点是发展微局部范畴的理论，例如拉格朗日子流形上支持的对象的全子范畴，并为微局部范畴发展一个合适的基础范畴，它的起源可能是拓扑的。该项目还旨在找到与船旗品种相关的微型地方类别的说明。进一步的研究是由量子化的想法推动的，量子化在这里是从物理学的概念抽象出来的几何术语。