Complex geometry and Toeplitz quantization

复杂几何和托普利茨量化

• 批准号: RGPIN-2016-03837
• 负责人: Barron, Tatyana
• 金额: $ 1.09万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614661
• 关键词: Complex; geometry; Toeplitz; quantization

Symplectic geometry provides a mathematical framework for classical mechanics. It enables a description of behavior of certain physical systems (for example, motion of a particle) in an elegant way. On a very small scale, in which equations of classical mechanics do not provide good results, laws of quantum mechanics serve as an appropriate replacement. Trying to understand the transition from classical mechanics to quantum mechanics leads to interesting mathematical problems. In this proposal some of these problems and possible approaches are described. On the classical side, I will study geometry of the phase space (for motion of a particle the phase space can be viewed as the space that parametrizes values of its position and velocity) and functions on this space (for example, energy of the particle). On the quantum side functions produce operators. In my research these operators are most often matrices. It is intuitively clear that everything is interconnected: geometry of the space, properties of functions and properties of matrices. My work will be concerned with establishing precise mathematical statements that describe these connections and how one affects the other. I will also develop approaches to creating these relationships (matrices from functions, or functions from geometry) for systems where existing techniques are not applicable and it is not immediately clear how to adapt them. Proposed research will provide contribution that will touch a number of areas of mathematics including differential geometry, semiclassical analysis, automorphic forms. Many of the projects have an underlying physical interpretation or motivation.

辛几何为经典力学提供了一个数学框架。它能够以优雅的方式描述某些物理系统的行为（例如，粒子的运动）。在非常小的尺度上，经典力学的方程不能提供好的结果，量子力学的定律可以作为适当的替代。试图理解从经典力学到量子力学的过渡会导致有趣的数学问题。在本提案中，描述了其中的一些问题和可能的方法。 在经典方面，我将研究相空间的几何（对于粒子的运动，相空间可以被看作是参数化其位置和速度值的空间）和在这个空间上的函数（例如，粒子的能量）。在量子方面，函数产生算子。在我的研究中，这些算子通常是矩阵。直观上很清楚，一切都是相互关联的：空间的几何，函数的性质和矩阵的性质。我的工作将是建立精确的数学陈述来描述这些联系以及一个如何影响另一个。我还将开发方法来创建这些关系（从函数矩阵，或从几何函数）的系统中，现有的技术是不适用的，它不是立即清楚如何适应它们。 拟议的研究将提供的贡献，将触及数学的一些领域，包括微分几何，半经典分析，自守形式。许多项目都有潜在的物理解释或动机。