Monoidal Categories and Topological Field Theory

Author: Vladimir Turaev,Alexis Virelizier

Publisher: Birkhäuser

ISBN: 3319498347

Category: Mathematics

Page: 523

View: 8528

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This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.
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Algebra, Arithmetic, and Geometry

Volume I: In Honor of Yu. I. Manin

Author: Yuri Tschinkel,Yuri Zarhin

Publisher: Springer Science & Business Media

ISBN: 9780817647452

Category: Mathematics

Page: 698

View: 7592

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EMAlgebra, Arithmetic, and Geometry: In Honor of Yu. I. ManinEM consists of invited expository and research articles on new developments arising from Manin’s outstanding contributions to mathematics.
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Mathematical Foundations of Quantum Field Theory and Perturbative String Theory

Author: Hisham Sati,Urs Schreiber

Publisher: American Mathematical Soc.

ISBN: 0821851950

Category: Mathematics

Page: 354

View: 1543

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Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory (QFT) has proven to be a rich source of ideas for mathematics for a long time. However, fundamental questions such as ``What is a QFT?'' did not have satisfactory mathematical answers, especially on spaces with arbitrary topology, fundamental for the formulation of perturbative string theory. This book contains a collection of papers highlighting the mathematical foundations of QFT and its relevance to perturbative string theory as well as the deep techniques that have been emerging in the last few years. The papers are organized under three main chapters: Foundations for Quantum Field Theory, Quantization of Field Theories, and Two-Dimensional Quantum Field Theories. An introduction, written by the editors, provides an overview of the main underlying themes that bind together the papers in the volume.
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Galois Theory, Hopf Algebras, and Semiabelian Categories

Author: George Janelidze

Publisher: American Mathematical Soc.

ISBN: 0821832905

Category: Mathematics

Page: 570

View: 6997

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This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabelian categories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers, specifically those interested in Galois theory and Hopf algebras and their categorical unification.
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Galois Theory, Hopf Algebras, and Semiabelian Categories

Author: George Janelidze, Bodo Pareigis, and Walter Tholen

Publisher: American Mathematical Soc.

ISBN: 9780821871478

Category: Mathematics

Page: N.A

View: 7578

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Survey and research papers in this volume are based on talks given at a workshop held at The Fields Institute for Research in the Mathematical Sciences (Toronto, ON, Canada). It provides an up-to-date account by leading researchers on the many current connections among Galois theories, Hopf algebras, and semiabelian categories.
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Mathematical Aspects of Quantum Field Theories

Author: Damien Calaque,Thomas Strobl

Publisher: Springer

ISBN: 3319099493

Category: Science

Page: 556

View: 7972

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Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras.
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Proceedings of the KirbyFest

papers dedicated to Rob Kirby on the occasion of his 60th birthday, 25 February 1998

Author: Joel Hass,Martin Scharlemann

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 575

View: 7861

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Fundamentals of Computation Theory

16th International Symposium, FCT 2007, Budapest, Hungary, August 27-30, 2007, Proceedings

Author: E. Csuhaj-Varjú

Publisher: Springer Science & Business Media

ISBN: 3540742395

Category: Computers

Page: 508

View: 2570

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This book constitutes the refereed proceedings of the 16th International Symposium Fundamentals of Computation Theory, FCT 2007, held in Budapest, Hungary in August 2007. The 39 revised full papers presented together with 4 invited papers were carefully reviewed and selected from 147 submissions. The papers address all current topics in computation theory such as automata and formal languages, design and analysis of algorithms, computational and structural complexity, semantics, logic, algebra and categories in computer science, circuits and networks, learning theory, specification and verification, parallel and distributed systems, concurrency theory, cryptography and cryptograhic protocols, approximation and randomized algorithms, computational geometry, quantum computation and information, bio-inspired computation.
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