Moduli Spaces of Riemann Surfaces

Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics.

Table of Contents

• A brief introduction to mapping class groups
• Teichmuller theory
• The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces
• Lectures on the Madsen-Weiss theorem
• The Torelli group and congruence subgroups of the mapping class group
• Tautological algebras of moduli spaces of curves
• Mirzakhani’s volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection
• numbers
• Teichmuller curves, mainly from the viewpoint of algebraic geometry
• Introduction to arithmetic mapping class groups