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Poisson Algebras of Holonomy Functions on Riemann Surfaces

黎曼曲面上完整函数的泊松代数

基本信息

批准号：
EP/J007234/1
负责人：
Marta Mazzocco
金额：
$ 14.25万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ007234%2F1
关键词：
Poisson Algebras Holonomy Functions Riemann

项目摘要

This is a project in Pure Mathematics (Integrable Systems), to attract two outstanding scientist, Prof. L. Chekhov, for one year and Prof. Jorgen Andersen for a total period of one month to the Mathematics Department at Loughborough University.Classically, physical phenomena are generally described by differential equations or, in other words, by equations which involve certain physical quantities (such as the position of a particle) and their variations (such as the particle velocity or its acceleration). Usually differential equations are very difficult or impossible to solve. Nevertheless there is a special class of differential equations (called integrable), which can be rewritten in the Lax form and therefore can be interpreted as an isospectral deformation. When we have a Lax representation for a physical system, then we can use many beautiful mathematical tools to understand, and often predict, its behaviour. In this project we will concentrate on a special class of equations which admit Lax representation: the so called Isomonodromic Deformations. In particular we will construct an isomonodromic deformation which will be related to a certain abstract algebra. Algebras of this kind give the correct set up for quantisation. Indeed, at quantum level the physical quantities are replaced by operators called observables belonging to some abstract algebras. For this reason the study of such algebras has many applications in Applied Mathematics and Theoretical Physics. Finally, we will give a geometric characterisation for this algebra, based on the celebrated Goldman bracket. This will allow us to establish a link between our work and the filed of Algebraic Geometry in Pure Mathematics.

这是纯数学（可积系统）的一个项目，吸引了两位杰出的科学家，L。契诃夫教授和Jorgen Andersen教授分别到拉夫堡大学数学系任教一年和一个月。传统上，物理现象通常用微分方程来描述，或者换句话说，用涉及某些物理量（如粒子的位置）及其变化（如粒子的速度或加速度）的方程来描述。通常微分方程是很难或不可能解决的。然而，有一类特殊的微分方程（称为可积的），它可以改写成Lax形式，因此可以解释为等谱变形。当我们有一个物理系统的Lax表示时，我们可以使用许多漂亮的数学工具来理解，并经常预测它的行为。在这个项目中，我们将集中在一类特殊的方程，承认拉克斯表示：所谓的Isomonodromic变形。特别地，我们将构造一个与某个抽象代数相关的等单径变形。这类代数给出了量化的正确设置。事实上，在量子层次上，物理量被称为观测量的算子所取代，这些观测量属于一些抽象代数。由于这个原因，研究这样的代数有许多应用在应用数学和理论物理。最后，我们将给出这个代数的几何特征，基于著名的高盛括号。这将使我们能够在我们的工作和纯数学中的代数几何领域之间建立联系。