Automorphic Lie Algebras - at the interface of mathematics and physics

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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.