# Lectures on Universal Teichmüller Space

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*Armen N. Sergeev*

A publication of the European Mathematical Society

This book is based on a lecture course given by the author
at the Educational Center of the Steklov Mathematical Institute in
2011. It is designed for a one-semester course for undergraduate
students familiar with basic differential geometry and complex and
functional analysis.

The universal Teichmüller space \(\mathcal{T}\) is the
quotient of the space of quasisymmetric homeomorphisms of the unit
circle modulo Möbius transformations. The first part of the book
is devoted to the study of geometric and analytic properties of
\(\mathcal{T}\). It is an infinite-dimensional Kähler
manifold which contains all classical Teichmüller spaces of
compact Riemann surfaces as complex submanifolds, which explains the
name “universal Teichmüller space”. Apart from classical
Teichmüller spaces, \(\mathcal{T}\) contains the space
\(\mathcal{S}\) of diffeomorphisms of the circle modulo
Möbius transformations. The latter space plays an important role
in the quantization of the theory of smooth strings.

The quantization of \(\mathcal{T}\) is presented in the
second part of the book. In contrast with the case of diffeomorphism
space \(\mathcal{S}\), which can be quantized in frames of the
conventional Dirac scheme, the quantization of \(\mathcal{T}\)
requires an absolutely different approach based on the noncommutative
geometry methods.

The book concludes with a list of 24 problems and exercises which
can used to prepare for examinations.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Undergraduate students familiar with basic differential geometry and complex and functional analysis.