Complex geometry and Toeplitz quantization

辛几何为经典力学提供了一个数学框架。它能够以一种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，经典力学方程不能提供好的结果，量子力学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转变导致了有趣的数学问题。在这个方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将其位置和速度的值参数化的空间)和在该空间上的函数(例如，粒子的能量)。在量子方面，函数产生运算符。在我的研究中，这些运算符通常是矩阵。很明显，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。我的工作将致力于建立精确的数学描述，描述这些联系以及其中一个如何影响另一个。我还将开发方法来为现有技术不适用且不清楚如何适应它们的系统创建这些关系(由函数生成的矩阵或由几何生成的函数)。*拟议的研究将提供涉及数学的多个领域，包括微分几何、半经典分析、自同构形式。许多项目都有潜在的物理解释或动机。*

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