Vladimir G. Turaev's Books

Homotopy Quantum Field Theory

By: Vladimir G. Turaev

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Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.

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300

Publisher

European Mathematical Society

Published Date

ISBN 10

3037190868

ISBN 13

9783037190869

Quantum Invariants of Knots and 3-Manifolds

By: Vladimir G. Turaev

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Quantum Invariants of Knots and 3-Manifolds

By: Vladimir G. Turaev

Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space. This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. Contents: Invariants of graphs in Euclidean 3-space and of closed 3-manifolds Foundations of topological quantum field theory Three-dimensional topological quantum field theory Two-dimensional modular functors 6j-symbols Simplicial state sums on 3-manifolds Shadows of manifolds and state sums on shadows Constructions of modular categories

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608

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110435225

ISBN 13

9783110435221

Topology, Ergodic Theory, Real Algebraic Geometry

Rokhlin's Memorial

By: Vladimir G. Turaev,Anatoliĭ Moiseevich Vershik

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Topology, Ergodic Theory, Real Algebraic Geometry

Rokhlin's Memorial

By: Vladimir G. Turaev,Anatoliĭ Moiseevich Vershik

This volume is dedicated to the memory of the Russian mathematician, V.A. Rokhlin (1919-1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmuller spaces, measure theory, etc. The book also includes a biography of Rokhlin by Vershik and two articles which should prove of historical interest.

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300

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821827405

ISBN 13

9780821827406

Book's by Vladimir G. Turaev

Homotopy Quantum Field Theory

Homotopy Quantum Field Theory

Quantum Invariants of Knots and 3-Manifolds

Quantum Invariants of Knots and 3-Manifolds

Topology, Ergodic Theory, Real Algebraic Geometry

Topology, Ergodic Theory, Real Algebraic Geometry

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