Braids, Links, and Mapping Class Groups
Braids, Links, and Mapping Class Groups

Author: Joan S. Birman

Publisher: Princeton University Press

Published: 1974

Total Pages: 244

ISBN-13: 9780691081496

DOWNLOAD EBOOK

The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.

Braids, Links, and Mapping Class Groups. (AM-82), Volume 82
Braids, Links, and Mapping Class Groups. (AM-82), Volume 82

Author: Joan S. Birman

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 241

ISBN-13: 1400881420

DOWNLOAD EBOOK

The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.

Geometry In Advanced Pure Mathematics
Geometry In Advanced Pure Mathematics

Author: Shaun Bullett

Publisher: World Scientific

Published: 2017-03-07

Total Pages: 235

ISBN-13: 1786341093

DOWNLOAD EBOOK

This book leads readers from a basic foundation to an advanced level understanding of geometry in advanced pure mathematics. Chapter by chapter, readers will be led from a foundation level understanding to advanced level understanding. This is the perfect text for graduate or PhD mathematical-science students looking for support in algebraic geometry, geometric group theory, modular group, holomorphic dynamics and hyperbolic geometry, syzygies and minimal resolutions, and minimal surfaces.Geometry in Advanced Pure Mathematics is the fourth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.

Combinatorial Group Theory
Combinatorial Group Theory

Author: Wilhelm Magnus

Publisher: Courier Corporation

Published: 2004-01-01

Total Pages: 466

ISBN-13: 0486438309

DOWNLOAD EBOOK

This seminal, much-cited account begins with a fairly elementary exposition of basic concepts and a discussion of factor groups and subgroups. The topics of Nielsen transformations, free and amalgamated products, and commutator calculus receive detailed treatment. The concluding chapter surveys word, conjugacy, and related problems; adjunction and embedding problems; and more. Second, revised 1976 edition.

Proceedings of the Rutgers Group Theory Year, 1983-1984
Proceedings of the Rutgers Group Theory Year, 1983-1984

Author: Michael Aschbacher

Publisher: Cambridge University Press

Published: 1984-12-28

Total Pages: 440

ISBN-13: 9780521264938

DOWNLOAD EBOOK

"With the classification of finite groups, an era of research in the subject ended. Some of the key figures in the classification program organized a research year at Rutgers University to analyze future directions of research in group theory. This volume is a record of the research year"- verso

The Mathematical Theory of Coding
The Mathematical Theory of Coding

Author: Ian F. Blake

Publisher: Academic Press

Published: 2014-05-10

Total Pages: 369

ISBN-13: 1483260593

DOWNLOAD EBOOK

The Mathematical Theory of Coding focuses on the application of algebraic and combinatoric methods to the coding theory, including linear transformations, vector spaces, and combinatorics. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on self-dual and quasicyclic codes, quadratic residues and codes, balanced incomplete block designs and codes, bounds on code dictionaries, code invariance under permutation groups, and linear transformations of vector spaces over finite fields. The text then takes a look at coding and combinatorics and the structure of semisimple rings. Topics include structure of cyclic codes and semisimple rings, group algebra and group characters, rings, ideals, and the minimum condition, chains and chain groups, dual chain groups, and matroids, graphs, and coding. The book ponders on group representations and group codes for the Gaussian channel, including distance properties of group codes, initial vector problem, modules, group algebras, andrepresentations, orthogonality relationships and properties of group characters, and representation of groups. The manuscript is a valuable source of data for mathematicians and researchers interested in the mathematical theory of coding.