Lectures on universal Teichmüller space (Q2863784)

This book presents a systematic description of universal Teichmüller spaces, their properties and applications; a notion which appeared in the work of L. Ahlfors and L. Bers on quasiconformal mappings.NEWLINENEWLINEThe book is written in the form of lecture notes and based on a cycle of lectures given by the author at the Steklov Institute in 2011. Anyway, the reader can consider each lecture as a special topic related to the main object of the book. The content of the book is related to the content of a previous book of the author [Geometric quantization of the loop space. Moskva: Matematicheskiĭ Institut im V. A. Steklova, RAN (2009)], in which the relationship of the space of diffeomorphisms \({\mathcal S}\) and the theory of strings was studied.NEWLINENEWLINEThe present book consists of fourteen lectures combined into 6 chapters. Chapter 1 ``Quasiconformal mappings'' has introductive character. It presents the notion of quasiconformal mapping and discusses its properties including existence and uniqueness and the case of quasisymmetric homeomorphisms. In Chapter 2 ``Universal Teichmüller space'' the above-mentioned space is introduced in different ways. Other parts of this chapter are related to the description of metric and topological properties of the universal Teichmüller space, to the construction of the Bers embedding, and to the existence of quasimetrics on this space. Chapter 3 ``Subspaces of the universal Teichmüller space'' is devoted to the study of different subspaces of the universal Teichmüller space, namely, to Riemann surfaces, classical Teichmüller spaces and the space of the normalized diffeomorphisms. The Grassmann realization of the universal Teichmüller space is discussed in Chapter 4. Chapter 5 deals with the quantization of the space of normalized diffeomorphisms. Starting from the notion of quantization of classical systems, the author comes later to the quantization of an extended system. In a sense it is preliminary for the final Chapter 6 ``Quantization of the universal Teichmüller space'' inspite the fact that for the later case it is necessary to use another scheme based on Connes' approach from noncommutative geometry.NEWLINENEWLINEIn the book, a high level of the mathematical theory is combined with a clear and straightforward presentation and very interesting explanations.