Automorphic Lie Algebras - at the interface of mathematics and physics

对非线性现象的研究，无论是在数学和物理的基础研究中，还是在应用中，都具有重要的意义。从物理学的角度来看，非线性方面在许多方面都发挥着相关的作用，从流体力学和非线性光学到力学，引力理论，量子场论和基本粒子理论。在数学方面，处理非线性问题需要发展基于代数、分析、几何和拓扑技术的新方法，并在所有这些领域取得新成果。非线性现象通常用微分方程来描述，这些微分方程通常很难或不可能求解。然而，有一类特殊的微分方程是可解的（在某种意义上）。它们被称为可积系统。当一个物理现象被一个可积系统描述时，它的行为就可以被全局地理解，并且通常可以被预测。现代数学物理中的许多概念，如孤子、瞬子、量子群等，都起源于可积系统理论。人们可以说，这个理论的美在于它的普适性：许多基本的非线性方程原来有一个普遍的性质，从而解释了显着的事实，即它们都是可积的和广泛的应用。近年来，可积系统的理论已重新制定的语言代数结构和许多新的数学对象被引入。这个公式是在纯数学、应用数学和理论物理学的许多学科的交叉点上。最近在这个框架中引入的新对象之一是一类新的代数，称为自守李代数，因为它们与自守函数的经典理论相似。这些代数无处不在，它们出现在数学和物理的许多分支中。因此，他们的研究是及时的，所获得的结果是广泛的科学界感兴趣的。最终目标是探索这些代数与可积系统理论和其他应用之间的联系。这项研究将开启以前未知的突破性研究。事实上，纯数学和应用数学分支之间的这种新联系将提出新的问题，并将证明意想不到的结果。

The study of nonlinear phenomena is of great importance both for fundamental research in mathematics and physics, and for applications. From the physical viewpoint nonlinear aspects play a relevant role in many contexts, from hydrodynamics and nonlinear optics to mechanics, gravitational theories, quantum field theories and the theory of elementary particles. On the mathematical side, treating nonlinearity requires the development of new methods based on algebraic, analytical, geometrical and topological techniques, and led to new results in all these areas. Nonlinear phenomena are generally described by differential equations which are usually very difficult or impossible to solve. Nevertheless there is a special class of differential equations which are solvable (in some sense). They are called integrable systems. When a physical phenomenon is described by an integrable system its behaviour can be understood globally and can be often predicted. Many concepts of modern mathematical physics such as solitons, instantons and quantum groups have their origin in theory of integrable systems. One may say that the beauty of this theory lies in its universality: many fundamental nonlinear equations turn out to have a universal character, thereby explaining the remarkable fact that they are both integrable and widely applicable.In recent years the theory of integrable systems has been reformulated in the language of algebraic structures and many new mathematical objects were introduced. This formulation lies at the crossroad of many disciplines in pure, applied mathematics and theoretical physics. One of the new objects introduced recently in this framework is a new class of algebras, called automorphic Lie algebras, for their analogy with the classical theory of automorphic functions. These algebras are ubiquitous, they appear in many branches of mathematics and physics. Their study is therefore timely and the results obtained are of interest for a wide scientific community. The ultimate goal is to explore the link between these algebras and the theory of integrable systems and other applications. This research will open previously unknown lines of ground breaking research. In fact, this new link between branches of pure and applied mathematics will pose new questions and unexpected results will be proven.