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Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Release : 2007
Category : Mathematics
ISBN : 3037191171

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Book Handbook of Teichmüller Theory Description/Summary:

For several decades, Teichmuller theory has been one of the most active research areas in mathematics, with a very wide range of points of view, including Riemann surface theory, hyperbolic geometry, low-dimensional topology, several complex variables, algebraic geometry, arithmetic, partial differential equations, dynamical systems, representation theory, symplectic geometry, geometric group theory, and mathematical physics. This book is the fourth volume in a Handbook of Teichmuller Theory project that started as an attempt to present, in a most comprehensive and systematic way, the various aspects of this theory with its relations to all the fields mentioned. The handbook is addressed to researchers as well as graduate students. This volume is divided into five parts: Part A: The metric and the analytic theory Part B: Representation theory and generalized structures Part C: Dynamics Part D: The quantum theory Part E: Sources Parts A, B, and D are sequels to parts on the same theme in previous volumes. Part E contains the translation together with a commentary of an important paper by Teichmuller that is almost unknown, even to specialists. Making the original ideas of and motivations for a theory clear is crucial for many reasons, and making this translation, together with the commentary that follows, available will give readers a broader perspective on Teichmuller theory. The various volumes in this collection are written by experts who have a broad view on the subject. In general, the chapters are expository, while some of them contain new and important results.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : European Mathematical Society
Release : 2007
Category : Mathematics
ISBN : 3037190558

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Book Handbook of Teichmüller Theory Description/Summary:

This multi-volume set deals with Teichmuller theory in the broadest sense, namely, as the study of moduli space of geometric structures on surfaces, with methods inspired or adapted from those of classical Teichmuller theory. The aim is to give a complete panorama of this generalized Teichmuller theory and of its applications in various fields of mathematics. The volumes consist of chapters, each of which is dedicated to a specific topic. The volume has 19 chapters and is divided into four parts: The metric and the analytic theory (uniformization, Weil-Petersson geometry, holomorphic families of Riemann surfaces, infinite-dimensional Teichmuller spaces, cohomology of moduli space, and the intersection theory of moduli space). The group theory (quasi-homomorphisms of mapping class groups, measurable rigidity of mapping class groups, applications to Lefschetz fibrations, affine groups of flat surfaces, braid groups, and Artin groups). Representation spaces and geometric structures (trace coordinates, invariant theory, complex projective structures, circle packings, and moduli spaces of Lorentz manifolds homeomorphic to the product of a surface with the real line). The Grothendieck-Teichmuller theory (dessins d'enfants, Grothendieck's reconstruction principle, and the Teichmuller theory of the solenoid). This handbook is an essential reference for graduate students and researchers interested in Teichmuller theory and its ramifications, in particular for mathematicians working in topology, geometry, algebraic geometry, dynamical systems and complex analysis. The authors are leading experts in the field.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Unknown
Release : 2020
Category : Teichmüller spaces
ISBN : 303719703X

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Book Handbook of Teichmüller Theory Description/Summary:

The present volume of the Handbook of Teichmüller theory is divided into three parts. The first part contains surveys on various topics in Teichmüller theory, including the complex structure of Teichmüller space, the Deligne-Mumford compactification of the moduli space, holomorphic quadratic differentials, Kleinian groups, hyperbolic 3-manifolds and the ending lamination theorem, the universal Teichmüller space, barycentric extensions of maps of the circle, and the theory of Higgs bundles. The second part consists of three historico-geometrical articles on Tissot (a precursor of the theory of quasiconfomal mappings), Grö̈tzsch and Lavrentieff, the two main founders of the modern theory of quasiconformal mappings. The third part comprises English translations of five papers by Grötzsch, a paper by Lavrentieff, and three papers by Teichmüller. These nine papers are foundational essays on the theories of conformal invariants and quasiconformal mappings, with applications to conformal geometry, to the type problem and to Nevanlinna's theory. The papers are followed by commentaries that highlight the relations between them and between later works on the subject. These papers are not only historical documents; they constitute an invaluable source of ideas for current research in Teichmüller theory.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Unknown
Release : 2022-06-29
Category : Uncategorized
ISBN : 303719555X

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Book Handbook of Teichmüller Theory Description/Summary:

This multi-volume set deals with Teichmüller theory in the broadest sense, namely, as the study of moduli space of geometric structures on surfaces, with methods inspired or adapted from those of classical Teichmüller theory. The aim is to give a complete panorama of this generalized Teichmüller theory and of its applications in various fields of mathematics. The volumes consist of chapters, each of which is dedicated to a specific topic. The present volume has 19 chapters and is divided into four parts: The metric and the analytic theory (uniformization, Weil-Petersson geometry, holomorphic families of Riemann surfaces, infinite-dimensional Teichmüller spaces, cohomology of moduli space, and the intersection theory of moduli space). The group theory (quasi-homomorphisms of mapping class groups, measurable rigidity of mapping class groups, applications to Lefschetz fibrations, affine groups of flat surfaces, braid groups, and Artin groups). Representation spaces and geometric structures (trace coordinates, invariant theory, complex projective structures, circle packings, and moduli spaces of Lorentz manifolds homeomorphic to the product of a surface with the real line). The Grothendieck-Teichmüller theory (dessins d'enfants, Grothendieck's reconstruction principle, and the Teichmüller theory of the soleniod). This handbook is an essential reference for graduate students and researchers interested in Teichmüller theory and its ramifications, in particular for mathematicians working in topology, geometry, algebraic geometry, dynamical systems and complex analysis. The authors are leading experts in the field.

Handbook of Homotopy Theory

Author : Haynes Miller
Publisher : CRC Press
Release : 2020-01-23
Category : Mathematics
ISBN : 9781351251617

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Book Handbook of Homotopy Theory Description/Summary:

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Unknown
Release : 2016
Category : MATHEMATICS
ISBN : 3037196610

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Book Handbook of Teichmüller Theory Description/Summary:

This volume is the sixth in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into five parts: Part A: The metric and the analytic theory. Part B: The group theory. Part C: Representation theory and generalized structures. Part D: The Grothendieck-Teichmüller theory. Part D: Sources. The topics surveyed include Grothendieck's construction of the analytic structure of Teichmüller space, identities on the geodesic length spectrum of hyperbolic surfaces (including Mirzakhani's application to the computation of Weil-Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d'enfants, and several others actions related to surfaces. The last part contains three papers by Teichmüller, translated into English with mathematical commentaries, and a document that contains H. Grötzsch's comments on Teichmüller's famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale. The handbook is addressed to researchers and to graduate students.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Unknown
Release : 2016
Category : Uncategorized
ISBN : 3037196602

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Book Handbook of Teichmüller Theory Description/Summary:

This volume is the fifth in a series dedicated to Teichmüller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts: Part A: The metric and the analytic theory Part B: The group theory Part C: Representation theory and generalized structures Part D: Sources The topics that are covered include identities for the hyperbolic geodesic length spectrum, Thurston's metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and there are many others. Besides surveying important parts of the theory, several chapters contain conjectures and open problems. The last part contains two fundamental papers by Teichmüller, translated into English and accompanied by mathematical commentaries. The chapters, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. They have an expository character (which fits with the original purpose of this handbook), but some of them also contain original and new material. The Handbook is addressed to researchers and to graduate students.

Handbook of Hilbert Geometry

Author : Athanase Papadopoulos,Marc Troyanov
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Release : 2014
Category : Mathematics
ISBN : 3037191473

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Book Handbook of Hilbert Geometry Description/Summary:

This volume presents surveys, written by experts in the field, on various classical and modern aspects of Hilbert geometry. They assume several points of view: Finsler geometry, calculus of variations, projective geometry, dynamical systems, and others. Some fruitful relations between Hilbert geometry and other subjects in mathematics are emphasized, including Teichmuller spaces, convexity theory, Perron-Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups and discrete group actions. This book is addressed to both students who want to learn the theory and researchers in this area.

Handbook of Teichmüller Theory

Author : Athanase Papadopoulos
Publisher : Unknown
Release : 2022-06-29
Category : Uncategorized
ISBN : 3037196033

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Book Handbook of Teichmüller Theory Description/Summary:

The subject of this handbook is Teichmüller theory in a wide sense, namely the theory of geometric structures on surfaces and their moduli spaces. This includes the study of vector bundles on these moduli spaces, the study of mapping class groups, the relation with 3-manifolds, the relation with symmetric spaces and arithmetic groups, the representation theory of fundamental groups, and applications to physics. Thus the handbook is a place where several fields of mathematics interact: Riemann surfaces, hyperbolic geometry, partial differential equations, several complex variables, algebraic geometry, algebraic topology, combinatorial topology, low-dimensional topology, theoretical physics, and others. This confluence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics. The present volume contains surveys on the fundamental theory as well as surveys on applications to and relations with the fields mentioned above. It is written by leading experts in the fields. Some of the surveys contain classical material, while others present the latest developments of the theory as well as open problems. The metric and the analytic theory. The group theory. The algebraic topology of mapping class groups and moduli spaces. Teichmüller theory and mathematical physics. The handbook is addressed to graduate students and researchers in all the fields mentioned.

Handbook of Complex Analysis

Author : Reiner Kuhnau
Publisher : Elsevier
Release : 2002-12-05
Category : Mathematics
ISBN : 0080532810

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Book Handbook of Complex Analysis Description/Summary:

Geometric Function Theory is a central part of Complex Analysis (one complex variable). The Handbook of Complex Analysis - Geometric Function Theory deals with this field and its many ramifications and relations to other areas of mathematics and physics. The theory of conformal and quasiconformal mappings plays a central role in this Handbook, for example a priori-estimates for these mappings which arise from solving extremal problems, and constructive methods are considered. As a new field the theory of circle packings which goes back to P. Koebe is included. The Handbook should be useful for experts as well as for mathematicians working in other areas, as well as for physicists and engineers. · A collection of independent survey articles in the field of GeometricFunction Theory · Existence theorems and qualitative properties of conformal and quasiconformal mappings · A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane)

Inverse Problem Theory and Methods for Model Parameter Estimation

Author : Albert Tarantola
Publisher : SIAM
Release : 2005-01-01
Category : Mathematics
ISBN : 0898717922

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Book Inverse Problem Theory and Methods for Model Parameter Estimation Description/Summary:

While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic.

Decorated Teichmüller Theory

Author : R. C. Penner
Publisher : European Mathematical Society
Release : 2012
Category : Mathematics
ISBN : 3037190752

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Book Decorated Teichmüller Theory Description/Summary:

There is an essentially ``tinker-toy'' model of a trivial bundle over the classical Teichmuller space of a punctured surface, called the decorated Teichmuller 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 algebro-geometric structure tied to the already elaborate combinatorial structure of the tinker-toy 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 Teichmuller 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.

Handbook of Analysis and Its Foundations

Author : Eric Schechter
Publisher : Academic Press
Release : 1996-10-24
Category : Mathematics
ISBN : 9780080532998

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Book Handbook of Analysis and Its Foundations Description/Summary:

Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. Covers some hard-to-find results including: Bessagas and Meyers converses of the Contraction Fixed Point Theorem Redefinition of subnets by Aarnes and Andenaes Ghermans characterization of topological convergences Neumanns nonlinear Closed Graph Theorem van Maarens geometry-free version of Sperners Lemma Includes a few advanced topics in functional analysis Features all areas of the foundations of analysis except geometry Combines material usually found in many different sources, making this unified treatment more convenient for the user Has its own webpage: http://math.vanderbilt.edu/

Lipman Bers, a Life in Mathematics

Author : Linda Keen,Irwin Kra,Rubí E. Rodríguez
Publisher : American Mathematical Soc.
Release : 2015-09-15
Category : Mathematicians
ISBN : 9781470420567

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Book Lipman Bers, a Life in Mathematics Description/Summary:

The book is part biography and part collection of mathematical essays that gives the reader a perspective on the evolution of an interesting mathematical life. It is all about Lipman Bers, a giant in the mathematical world who lived in turbulent and exciting times. It captures the essence of his mathematics, a development and transition from applied mathematics to complex analysis--quasiconformal mappings and moduli of Riemann surfaces--and the essence of his personality, a progression from a young revolutionary refugee to an elder statesman in the world of mathematics and a fighter for global human rights and the end of political torture. The book contains autobiographical material and short reprints of his work. The main content is in the exposition of his research contributions, sometimes with novel points of view, by students, grand-students, and colleagues. The research described was fundamental to the growth of a central part of 20th century mathematics that, now in the 21st century, is in a healthy state with much current interest and activity. The addition of personal recollections, professional tributes, and photographs yields a picture of a man, his personal and professional family, and his time.

Moduli Spaces of Riemann Surfaces

Author : Benson Farb,Richard Hain,Eduard Looijenga
Publisher : American Mathematical Soc.
Release : 2013-08-16
Category : Mathematics
ISBN : 9780821898871

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Book Moduli Spaces of Riemann Surfaces Description/Summary:

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.

In the Tradition of Thurston

Author : Ken’ichi Ohshika,Athanase Papadopoulos
Publisher : Springer Nature
Release : 2021-01-08
Category : Mathematics
ISBN : 9783030559281

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Book In the Tradition of Thurston Description/Summary:

This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups, foliations, mapping class groups, Teichmüller theory, anti-de Sitter geometry, and co-Minkowski geometry. The book is addressed to researchers and students who want to learn about Thurston’s wide-ranging mathematical ideas and their impact. At the same time, it is a tribute to Thurston, one of the greatest geometers of all time, whose work extended over many fields in mathematics and who had a unique way of perceiving forms and patterns, and of communicating and writing mathematics.