Complex geometry and Toeplitz quantization

复杂几何和托普利茨量化

基本信息

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：
2018
资助国家：
加拿大
起止时间：
2018-01-01 至 2019-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=644717
关键词：
Complex geometry Toeplitz quantization

项目摘要

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. *** *** *** **

符号几何形状为经典力学提供了数学框架。它可以以优雅的方式描述某些物理系统的行为（例如粒子的运动）。在非常小的规模上，经典力学方程不提供良好的结果，量子力学定律是适当的替代品。试图了解从经典力学到量子力学的过渡会导致有趣的数学问题。在此提案中，描述了一些问题和可能的方法。在量子侧功能上会产生操作员。在我的研究中，这些操作员通常是矩阵。直观上清楚的是，所有内容都是互连的：空间的几何形状，函数的特性和矩阵的特性。我的工作将关注建立描述这些联系以及一个人如何影响另一个联系的精确数学陈述。我还将开发用于创建这些关系（函数的矩阵或几何功能的矩阵）的方法，这些系统不适用现有技术，并且尚不清楚如何适应它们。 ***拟议的研究将提供贡献，该贡献将触及许多数学领域，包括差异几何，半经典分析，自动形式。许多项目都有基本的物理解释或动力。 *** *** *** **