The QGM Master Class Series 2012; 377 pp; hardcover Volume: 1 ISBN10: 3037190752 ISBN13: 9783037190753 List Price: US$78 Member Price: US$62.40 Order Code: EMSQGM/1
 There is an essentially "tinkertoy" model of a trivial bundle over the classical Teichmüller space of a punctured surface, called the decorated Teichmüller space, where the fiber over a point is the space of all tuples of horocycles, one about each puncture. This model leads to an extension of the classical mapping class groups called the Ptolemy groupoids and to certain matrix models solving related enumerative problems, each of which has proved useful both in mathematics and in theoretical physics. These spaces enjoy several related parametrizations leading to a rich and intricate algebrogeometric structure tied to the already elaborate combinatorial structure of the tinkertoy model. Indeed, the natural coordinates give the prototypical examples not only of cluster algebras but also of tropicalization. This interplay of combinatorics and coordinates admits further manifestations, for example, in a Lie theory for homeomorphisms of the circle, in the geometry underlying the Gauss product, in profinite and pronilpotent geometry, in the combinatorics underlying conformal and topological quantum field theories, and in the geometry and combinatorics of macromolecules. This volume gives the story a wider context of these decorated Teichmüller spaces as developed by the author over the last two decades in a series of papers, some of them in collaboration. Sometimes correcting errors or typos, sometimes simplifying proofs, and sometimes articulating more general formulations than the original research papers, this volume is self contained and requires little formal background. Based on a master's course at Aarhus University, it gives the first treatment of these works in monographic form. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in the study of geometrical aspects and mathematical foundations of quantum field theory and string theory. Table of Contents  The basics
 Lambda lengths in finite dimensions
 Lambda lengths in infinite dimensions
 Decomposition of the decorated spaces
 Mapping class groupoids and moduli spaces
 Further applications
 Epilogue
 Appendix A. Geometry of Gauss product
 Appendix B. Dual to the Kähler two form
 Appendix C. Stable curves and screens
 Bibliography
 List of notation
 Index
