Frobenius Algebras and 2-D Topological Quantum Field Theories

Author: Joachim Kock

Publisher: Cambridge University Press

ISBN: 9780521540315

Category: Mathematics

Page: 240

View: 7330

This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.

Deep Beauty

Understanding the Quantum World through Mathematical Innovation

Author: Hans Halvorson

Publisher: Cambridge University Press

ISBN: 113949922X

Category: Mathematics

Page: N.A

View: 4655

No scientific theory has caused more puzzlement and confusion than quantum theory. Physics is supposed to help us to understand the world, but quantum theory makes it seem a very strange place. This book is about how mathematical innovation can help us gain deeper insight into the structure of the physical world. Chapters by top researchers in the mathematical foundations of physics explore new ideas, especially novel mathematical concepts at the cutting edge of future physics. These creative developments in mathematics may catalyze the advances that enable us to understand our current physical theories, especially quantum theory. The authors bring diverse perspectives, unified only by the attempt to introduce fresh concepts that will open up new vistas in our understanding of future physics.

Monoidal Categories and Topological Field Theory

Author: Vladimir Turaev,Alexis Virelizier

Publisher: Birkhäuser

ISBN: 3319498347

Category: Mathematics

Page: 523

View: 6325

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.

Topology and Field Theories

Author: Stephan Stolz

Publisher: American Mathematical Soc.

ISBN: 147041015X

Category: Mathematics

Page: 176

View: 4419

This book is a collection of expository articles based on four lecture series presented during the 2012 Notre Dame Summer School in Topology and Field Theories. The four topics covered in this volume are: Construction of a local conformal field theory associated to a compact Lie group, a level and a Frobenius object in the corresponding fusion category; Field theory interpretation of certain polynomial invariants associated to knots and links; Homotopy theoretic construction of far-reaching generalizations of the topological field theories that Dijkgraf and Witten associated to finite groups; and a discussion of the action of the orthogonal group on the full subcategory of an -category consisting of the fully dualizable objects. The expository style of the articles enables non-experts to understand the basic ideas of this wide range of important topics.

Stacks and Categories in Geometry, Topology, and Algebra

Author: Tony Pantev,Carlos Simpson,Bertrand Toën,Michel Vaquié,Gabriele Vezzosi

Publisher: American Mathematical Soc.

ISBN: 1470415577

Category: Algebra

Page: 323

View: 1544

This volume contains the proceedings of the CATS4 Conference on Higher Categorical Structures and their Interactions with Algebraic Geometry, Algebraic Topology and Algebra, held from July 2-7, 2012, at CIRM in Luminy, France. Over the past several years, the CATS conference series has brought together top level researchers from around the world interested in relative and higher category theory and its applications to classical mathematical domains. Included in this volume is a collection of articles covering the applications of categories and stacks to geometry, topology and algebra. Techniques such as localization, model categories, simplicial objects, sheaves of categories, mapping stacks, dg structures, hereditary categories, and derived stacks, are applied to give new insight on cluster algebra, Lagrangians, trace theories, loop spaces, structured surfaces, stability, ind-coherent complexes and 1-affineness showing up in geometric Langlands, branching out to many related topics along the way.

Completely Bounded Maps and Operator Algebras

Author: Vern Paulsen

Publisher: Cambridge University Press

ISBN: 9780521816694

Category: Mathematics

Page: 300

View: 7963

In this book, first published in 2003, the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensable introduction to the theory of operator spaces for all who want to know more.

Tensor Categories

Author: Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik

Publisher: American Mathematical Soc.

ISBN: 1470434415

Category:

Page: 344

View: 2704

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Knot Invariants and Higher Representation Theory

Author: Ben Webster

Publisher: American Mathematical Soc.

ISBN: 1470426501

Category: Functor theory

Page: 141

View: 6835

The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for sl and sl and by Mazorchuk-Stroppel and Sussan for sl . The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl , the author shows that these categories agree with certain subcategories of parabolic category for gl .

Basic Noncommutative Geometry

Author: Masoud Khalkhali

Publisher: European Mathematical Society

ISBN: 9783037190616

Category: Mathematics

Page: 223

View: 2707

"Basic Noncommutative Geometry provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well."--Publisher's description.

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 9314

New Examples of Frobenius Extensions

Author: Lars Kadison

Publisher: American Mathematical Soc.

ISBN: 0821819623

Category: Mathematics

Page: 84

View: 9791

The 14th volume in an inexpensive softcover series designed for students especially.

The Development of Mathematics Throughout the Centuries

A Brief History in a Cultural Context

Author: Brian Evans

Publisher: John Wiley & Sons

ISBN: 1118853970

Category: Mathematics

Page: 252

View: 2008

Throughout the book, readers take a journey throughout time and observe how people around the world have understood these patterns of quantity, structure, and dimension around them. The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Contex provides a brief overview of the history of mathematics in a very straightforward and understandable manner and also addresses major findings that influenced the development of mathematics as a coherent discipline. This book: Highlights the contributions made by various world cultures including African, Egyptian, Babylonian, Chinese, Indian, Islamic, and pre-Columbian American mathematics Features an approach that is not too rigorous and is ideal for a one-semester course of the history of mathematics. Includes a Resources and Recommended Reading section for further exploration and has been extensively classroom-tested

A Taste of Jordan Algebras

Author: Kevin McCrimmon

Publisher: Springer Science & Business Media

ISBN: 0387217967

Category: Mathematics

Page: 563

View: 8787

This book describes the history of Jordan algebras and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.

Matter and Motion

Author: James Clerk Maxwell

Publisher: N.A

ISBN: N.A

Category: Force and energy

Page: 224

View: 4642

The British National Bibliography

Author: Arthur James Wells

Publisher: N.A

ISBN: N.A

Category: English literature

Page: N.A

View: 891

Towards Higher Categories

Author: John C. Baez,J. Peter May

Publisher: Springer Science & Business Media

ISBN: 1441915362

Category:

Page: 292

View: 8211

Extremes and Recurrence in Dynamical Systems

Author: Valerio Lucarini,Ana Cristina Gomes Monteiro Moreira de Freitas,Davide Faranda,Jorge Milhazes Freitas,Mark Holland,Tobias Kuna,Matthew Nicol,Mike Todd,Sandro Vaienti

Publisher: John Wiley & Sons

ISBN: 1118632192

Category: Mathematics

Page: 312

View: 2121

Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.