Automorphic Lie Algebras - at the interface of mathematics and physics

非线性现象的研究对于数学和物理的基础研究以及应用都具有重要意义。从物理学的角度来看，非线性方面在许多领域发挥着相关的作用，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，处理非线性需要发展基于代数、解析、几何和拓扑技术的新方法，并在所有这些领域产生新的结果。非线性现象通常用微分方程来描述，而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。当一个物理现象用一个可积系统来描述时，它的行为就可以被全面地理解，而且常常可以被预测。现代数学物理中的许多概念，如孤子、瞬子和量子群，都起源于可积系统理论。有人可能会说，这个理论的美妙之处在于它的通用性：许多基本的非线性方程被证明具有通用性，从而解释了它们既可积又广泛适用的显著事实。近年来，可积系统的理论被用代数结构的语言重新表述，并引入了许多新的数学对象。这个公式是纯数学、应用数学和理论物理中许多学科的交叉点。最近在这个框架中引入的一个新对象是一类新的代数，称为自同构李代数，因为它们与经典的自同构函数理论相似。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，所获得的结果引起了广泛科学界的兴趣。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。这项研究将开辟以前未知的突破性研究领域。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，意想不到的结果将得到证明。

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.