Poisson Algebras of Holonomy Functions on Riemann Surfaces

这是纯粹数学(可积系统)的一个项目，旨在吸引两位杰出的科学家L·契诃夫教授和约尔根·安德森教授来拉夫堡大学数学系工作一年和约尔根·安德森教授总共一个月。经典的物理现象通常用微分方程来描述，换句话说，用涉及某些物理量(如粒子的位置)及其变化(如粒子速度或其加速度)的方程来描述。通常，微分方程式是很难或不可能解的。然而，有一类特殊的微分方程(称为可积方程)可以改写为Lax形式，因此可以解释为等谱变形。当我们对一个物理系统有一个松散的表示时，我们就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，我们将集中讨论一类允许松散表示的特殊方程：所谓的等单行变形。特别地，我们将构造一个与某个抽象代数相关的同素形变。这类代数给出了正确的量子化设置。事实上，在量子水平上，物理量被属于某些抽象代数的称为可观测的运算符所取代。因此，对这类代数的研究在应用数学和理论物理中有着广泛的应用。最后，我们将以著名的高盛为基础，给出这个代数的几何特征。这将允许我们在我们的工作和纯数学中的代数几何领域建立联系。

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.