Integrable systems, gravity and moduli spaces

可积系统的现代理论可以追溯到19世纪世纪，当时英国工程师约翰·斯科特·罗素在埃米斯顿的联合运河中观察到了一个孤子（他称之为“平移波”）。这一事件发生在1834年8月，直到世纪中期，现代可积系统理论（特别是罗素提出的描述浅水非线性波的Korteveg-de-弗里斯方程理论）开始爆炸性发展时，才得到了解释。其他根源的理论在于经典几何曲面：另一个经典的可积系统-正弦戈登方程-出现在作品的比安奇在20世纪20年代;大约在同一时间的理论isomonodromy变形的施莱辛格和Painlevé方程被发现。到目前为止，可积系统理论已经发展成为数学和物理学各个分支的一个非常有力的工具。 可积系统演化的一个基本发展发生在80年代，当时日本的神保学派将两个遥远的科学领域联系起来;一方面是可积波理论和统计物理学，另一方面是对常微分方程分类感兴趣的P. Painlevé的纯数学奋进。这一合并的结果是发明的理论的某些特殊类的职能，命名为“τ函数”，现在出现在越来越多的领域的数学，物理学，甚至组合。该项目的目标是发展理论τ函数和应用到模空间的理论，某些动力系统介绍了N。希钦在90年代和理论的薛定谔方程的黎曼曲面和它的“半经典”分析。 我们将定义非平凡拓扑的Riemann曲面（“高亏格曲面”）上的等单值变形的τ函数，并将其与单值映射的Hamilton形式联系起来。此外，我们将把τ-函数理论应用于希钦系统。我们还将研究黎曼曲面上的薛迪格方程，以及它的WKB近似和单值映射。这个物体有着丰富而美丽的几何结构，在杨-米尔斯方程理论中也起着重要的作用。 经典可积系统的自然量子化产生了许多物理上重要的量子模型，它们可以被显式地处理。其中一个模型出现在引力理论中-它是具有两个偏振的爱因斯坦-罗森波模型。这样的模型可以在代数水平上量子化，并且它被证明是量子化全四维引力的自然方法之一。 完成非线性爱因斯坦-罗森波量子模型的开发是该提案的主要目标之一。

The modern theory of integrable systems can be traced back to the XIXth century, when British engineer John Scott Russell observed a soliton (which he called the "wave of translation") in the Union Canal at Hermiston, Edinbourg. This event, which happened in August, 1834, found its explanation only in the middle of 20th century, when the modern theory of integrable systems (and in particular the theory of Korteveg-de-Vries equation which describes non-linear waves on shallow water observed by Russell) started its explosive development. Other roots of the theory lie in the classical geometry of surfaces: another classical integrable system - the sine-Gordon equation - appeared in works of Bianchi in 1920's; about the same time the theory of isomonodromy deformations by Schlesinger and Painlevé equations were discovered. By now the theory of integrable systems has developed into a very powerful tool used in various branches of mathematics and physics. A fundamental development in the evolution of integrable systems occurred in the '80s when the Japanese school of Jimbo connected two distant areas of science; the theory of integrable waves and statistical physics on one side and the pure mathematical endeavor of P. Painlevé, who was interested in the classification of ordinary differential equations. The result of this merge was the invention of the theory of a certain special class of functions, named "tau-functions" which now appear in ever more numerous areas of mathematics, physics and even combinatorics. The goal of the project is to develop the theory tau functions and apply to the theory of moduli spaces, certain dynamical systems introduced by N. Hitchin in the '90s and the theory of the Schrödinger equation on Riemann surfaces and its "semiclassical" analysis. We are going to define the tau-function for isomonodromic deformations on Riemann surfaces of non-trivial topology (the "higher genus surfaces") and relate it to the Hamiltonian formalism of monodromy map. Furthermore, we are going to apply the theory of tau-functions to Hitchin systems. We shall also study the Schrödiger equation on a Riemann surface, together with its Wentzel-Kramers-Brillouin (WKB) approximation and monodromy map. This object turns out to have a rich and beautiful geometric structure, and also plays an important role in the theory of Yang-Mills equations. Natural quantization of classical integrable systems produces many physically important quantum models which can be treated explicitly. One of such models appears in the theory of gravity - it is the model of Einstein-Rosen waves with two polarizations. Such model can be quantized on algebraic level, and it turns out to arise in one of natural approaches to quantization of the full four-dimensional gravity. The completion of the development of the quantum model of non-linear Einstein-Rosen waves is one of the main goals of the proposal.