Weil-Petersson Metric on the Universal Teichmuller Space

Weil-Petersson Metric on the Universal Teichmuller Space
Title Weil-Petersson Metric on the Universal Teichmuller Space PDF eBook
Author Leon Armenovich Takhtadzhi︠a︡n
Publisher American Mathematical Soc.
Pages 136
Release 2006
Genre Mathematics
ISBN 0821839365

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In this memoir, we prove that the universal Teichmuller space $T(1)$ carries a new structure of a complex Hilbert manifold and show that the connected component of the identity of $T(1)$ -- the Hilbert submanifold $T {0 (1)$ -- is a topological group. We define a Weil-Petersson metric on $T(1)$ by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that $T(1)$ is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmuller curve fibration over the universal Teichmuller space. As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller space. We study in detail the Hilbert manifold structure on $T {0 (1)$ and characterize points on $T {0 (1)$ in terms of Bers and pre-Bers embeddings by proving that the Grunsky operators $B {1 $ and The results of this memoir were presented in our e-prints: Weil-Petersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004).

Moduli Spaces of Riemann Surfaces

Moduli Spaces of Riemann Surfaces
Title Moduli Spaces of Riemann Surfaces PDF eBook
Author Benson Farb
Publisher American Mathematical Soc.
Pages 371
Release 2013-08-16
Genre Mathematics
ISBN 0821898876

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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. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

Teichmüller Theory and Applications to Geometry, Topology, and Dynamics
Title Teichmüller Theory and Applications to Geometry, Topology, and Dynamics PDF eBook
Author John Hamal Hubbard
Publisher
Pages 576
Release 2022-02
Genre
ISBN 9781943863013

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In the Tradition of Thurston III

In the Tradition of Thurston III
Title In the Tradition of Thurston III PDF eBook
Author Ken’ichi Ohshika
Publisher Springer Nature
Pages 456
Release
Genre
ISBN 3031435028

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Problems on Mapping Class Groups and Related Topics

Problems on Mapping Class Groups and Related Topics
Title Problems on Mapping Class Groups and Related Topics PDF eBook
Author Benson Farb
Publisher American Mathematical Soc.
Pages 384
Release 2006-09-12
Genre Mathematics
ISBN 0821838385

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The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.

Lie Algebras, Vertex Operator Algebras, and Related Topics

Lie Algebras, Vertex Operator Algebras, and Related Topics
Title Lie Algebras, Vertex Operator Algebras, and Related Topics PDF eBook
Author Katrina Barron
Publisher American Mathematical Soc.
Pages 282
Release 2017-08-15
Genre Mathematics
ISBN 1470426668

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This volume contains the proceedings of the conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, celebrating the 70th birthday of James Lepowsky and Robert Wilson, held from August 14–18, 2015, at the University of Notre Dame, Notre Dame, Indiana. Since their seminal work in the 1970s, Lepowsky and Wilson, their collaborators, their students, and those inspired by their work, have developed an amazing body of work intertwining the fields of Lie algebras, vertex algebras, number theory, theoretical physics, quantum groups, the representation theory of finite simple groups, and more. The papers presented here include recent results and descriptions of ongoing research initiatives representing the broad influence and deep connections brought about by the work of Lepowsky and Wilson and include a contribution by Yi-Zhi Huang summarizing some major open problems in these areas, in particular as they pertain to two-dimensional conformal field theory.

Hamiltonian Reduction by Stages

Hamiltonian Reduction by Stages
Title Hamiltonian Reduction by Stages PDF eBook
Author Jerrold E. Marsden
Publisher Springer
Pages 527
Release 2007-06-05
Genre Mathematics
ISBN 3540724702

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This volume provides a detailed account of the theory of symplectic reduction by stages, along with numerous illustrations of the theory. It gives special emphasis to group extensions, including a detailed discussion of the Euclidean group, the oscillator group, the Bott-Virasoro group and other groups of matrices. The volume also provides ample background theory on symplectic reduction and cotangent bundle reduction.