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Author: Benson Farb Publisher: Princeton University Press ISBN: 0691147949 Category : Mathematics Languages : en Pages : 490
Book Description
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
Author: Benson Farb Publisher: Princeton University Press ISBN: 0691147949 Category : Mathematics Languages : en Pages : 490
Book Description
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
Author: Benson Farb Publisher: American Mathematical Soc. ISBN: 0821838385 Category : Class groups (Mathematics) Languages : en Pages : 384
Book Description
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.
Author: Matt Clay Publisher: Princeton University Press ISBN: 1400885396 Category : Mathematics Languages : en Pages : 456
Book Description
Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.
Author: Richard Evan Schwartz Publisher: American Mathematical Soc. ISBN: 0821853686 Category : Hypersurfaces Languages : en Pages : 330
Book Description
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
Author: Benson Farb Publisher: American Mathematical Soc. ISBN: 0821898876 Category : Mathematics Languages : en Pages : 371
Book Description
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.
Author: J. P. May Publisher: University of Chicago Press ISBN: 9780226511832 Category : Mathematics Languages : en Pages : 262
Book Description
Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
Author: Sen Hu Publisher: World Scientific ISBN: 9810239092 Category : Gauge fields (Physics). Languages : en Pages : 214
Book Description
This monograph is based on lectures on topological quantum field theory given in 1989 at Princeton University by E. Witten, in which he unified several important mathematical works in terms of the Donaldson polynomial, Gromov/Floer homology, and Jones polynomials. Witten explained his three-dimensional construction of Jones polynomials, "an elegant construction of a new polynomial invariant in three-dimensional space" (per the author), via quantization of Chern-Simons gauge theory. Hu (Princeton U.) adds missing details and some new developments in the field. Annotation copyrighted by Book News Inc., Portland, OR.
Author: Albert Fathi Publisher: Princeton University Press ISBN: 1400839033 Category : Mathematics Languages : en Pages : 272
Book Description
This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo-Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory. Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology. Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.
Author: Benson Farb Publisher: ISBN: Category : Class groups (Mathematics) Languages : en Pages : 472
Book Description
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.The book begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmoller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification"--Provided by publisher
Author: Benson Farb
Author: Benson Farb
Author: Benson Farb
Author: Matt Clay
Author: Richard Evan Schwartz
Author: Benson Farb
Author: Sebastian Baader
Author: J. P. May
Author: Sen Hu
Author: Albert Fathi
Author: Benson Farb