• Mathematics

Frobenius Algebras and 2-D Topological Quantum Field Theories


Author: Joachim Kock
Publisher: Cambridge University Press
ISBN: 9780521540315
Category: Mathematics
Page: 240
View: 2812
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.

    • Mathematics

Deep Beauty

Understanding the Quantum World through Mathematical Innovation
Author: Hans Halvorson
Publisher: Cambridge University Press
ISBN: 113949922X
Category: Mathematics
Page: N.A
View: 472
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.

    • Mathematics

Monoidal Categories and Topological Field Theory


Author: Vladimir Turaev,Alexis Virelizier
Publisher: Birkhäuser
ISBN: 3319498347
Category: Mathematics
Page: 523
View: 1734
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.

    • Mathematics

Topology and Field Theories


Author: Stephan Stolz
Publisher: American Mathematical Soc.
ISBN: 147041015X
Category: Mathematics
Page: 176
View: 4877
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.

    • Algebra

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: 4539
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.


    • Mathematics

New Examples of Frobenius Extensions


Author: Lars Kadison
Publisher: American Mathematical Soc.
ISBN: 0821819623
Category: Mathematics
Page: 84
View: 3967
The 14th volume in an inexpensive softcover series designed for students especially.

    • Mathematics

Mathematical Reviews


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 7222

    • Mathematics

Algebraic Operads


Author: Jean-Louis Loday,Bruno Vallette
Publisher: Springer Science & Business Media
ISBN: 3642303625
Category: Mathematics
Page: 636
View: 1341
In many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.

    • Science

Homotopy Quantum Field Theory


Author: Vladimir G. Turaev
Publisher: European Mathematical Society
ISBN: 9783037190869
Category: Science
Page: 276
View: 2764

    • Mathematics

2016 MATRIX Annals


Author: Jan de Gier,Cheryl E. Praeger,Terence Tao
Publisher: Springer
ISBN: 3319722999
Category: Mathematics
Page: 656
View: 9521
MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: Higher Structures in Geometry and Physics (Chapters 1-5 and 18-21); Winter of Disconnectedness (Chapter 6 and 22-26); Approximation and Optimisation (Chapters 7-8); Refining C*-Algebraic Invariants for Dynamics using KK-theory (Chapters 9-13); Interactions between Topological Recursion, Modularity, Quantum Invariants and Low-dimensional Topology (Chapters 14-17 and 27). The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.

    • Science

Mathematical Analysis of Evolution, Information, and Complexity


Author: Wolfgang Arendt,Wolfgang P. Schleich
Publisher: John Wiley & Sons
ISBN: 3527628037
Category: Science
Page: 502
View: 740
Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.

    • Science

Elements for Physics

Quantities, Qualities, and Intrinsic Theories
Author: Albert Tarantola
Publisher: Springer Science & Business Media
ISBN: 3540311076
Category: Science
Page: 266
View: 2162
Reviews and extends the theory of Lie groups, develops differential geometry, proposing compact definitions of torsion and of curvature, and adapts the usual notion of linear tangent application to the intrinsic point of view proposed for physics. Uses a unifying illustration: two simple theories are studied with some detail, the theory of heat conduction and the theory of linear elastic media. Shows that the resulting equations derived in this manner differ quantitatively and qualitatively from those usually presented.

    • Algebra, Homological

Colored Operads


Author: Donald Yau
Publisher: American Mathematical Soc.
ISBN: 1470427230
Category: Algebra, Homological
Page: 428
View: 4684
The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality. The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.

    • English literature

The British National Bibliography


Author: Arthur James Wells
Publisher: N.A
ISBN: N.A
Category: English literature
Page: N.A
View: 7025

    • Mathematics

A Taste of Jordan Algebras


Author: Kevin McCrimmon
Publisher: Springer Science & Business Media
ISBN: 0387217967
Category: Mathematics
Page: 563
View: 7852
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.

    • Mathematics

Motives, Quantum Field Theory, and Pseudodifferential Operators

Conference on Motives, Quantum Field Theory, and Pseudodifferential Operators, June 2-13, 2008, Boston University, Boston, Massachusetts
Author: Alan L. Carey
Publisher: American Mathematical Soc.
ISBN: 0821851993
Category: Mathematics
Page: 349
View: 7993
This volume contains articles related to the conference ``Motives, Quantum Field Theory, and Pseudodifferntial Operators'' held at Boston University in June 2008, with partial support from the Clay Mathematics Institute, Boston University, and the National Science Foundation. There are deep but only partially understood connections between the three conference fields, so this book is intended both to explain the known connections and to offer directions for further research. In keeping with the organization of the conference, this book contains introductory lectures on each of the conference themes and research articles on current topics in these fields. The introductory lectures are suitable for graduate students and new Ph.D.'s in both mathematics and theoretical physics, as well as for senior researchers, since few mathematicians are expert in any two of the conference areas. Among the topics discussed in the introductory lectures are the appearance of multiple zeta values both as periods of motives and in Feynman integral calculations in perturbative QFT, the use of Hopf algebra techniques for renormalization in QFT, and regularized traces of pseudodifferential operators. The motivic interpretation of multiple zeta values points to a fundamental link between motives and QFT, and there are strong parallels between regularized traces and Feynman integral techniques. The research articles cover a range of topics in areas related to the conference themes, including geometric, Hopf algebraic, analytic, motivic and computational aspects of quantum field theory and mirror symmetry. There is no unifying theory of the conference areas at present, so the research articles present the current state of the art pointing towards such a unification.

Tensor Categories


Author: Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher: American Mathematical Soc.
ISBN: 1470434415
Category:
Page: 344
View: 6893
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.

    • Mathematics

Mirror Symmetry and Algebraic Geometry


Author: David A. Cox,Sheldon Katz
Publisher: American Mathematical Soc.
ISBN: 082182127X
Category: Mathematics
Page: 469
View: 2321
Mathematicians wanting to get into the field ... will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. --Bulletin of the LMS The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. --Mathematical Reviews Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is a completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

    • Mathematics

Higher Structures in Geometry and Physics

In Honor of Murray Gerstenhaber and Jim Stasheff
Author: Alberto S. Cattaneo,Anthony Giaquinto,Ping Xu
Publisher: Springer Science & Business Media
ISBN: 9780817647353
Category: Mathematics
Page: 362
View: 751
This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. The volume is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures.