An Excursion in Diagrammatic Algebra

An Excursion in Diagrammatic Algebra
Author: J. Scott Carter
Publisher: World Scientific
Total Pages: 294
Release: 2012
Genre: Mathematics
ISBN: 9814374504
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1. A sphere -- 2. Surfaces, folds, and cusps -- 3. The inside and outside -- 4. Dimensions -- 5. Immersed surfaces -- 6. Movies -- 7. Movie moves -- 8. Taxonomic summary -- 9. How not to turn the sphere inside-out -- 10. A physical metaphor -- 11. Sarah's thesis -- 12. The eversion -- 13. The double point and fold surfaces

An Excursion in Diagrammatic Algebra

An Excursion in Diagrammatic Algebra
Author: J. Scott Carter
Publisher: World Scientific
Total Pages: 294
Release: 2012
Genre: Mathematics
ISBN: 9814374490
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The aim of this book is to give as detailed a description as is possible of one of the most beautiful and complicated examples in low-dimensional topology. This example is a gateway to a new idea of higher dimensional algebra in which diagrams replace algebraic expressions and relationships between diagrams represent algebraic relations. The reader may examine the changes in the illustrations in a leisurely fashion; or with scrutiny, the reader will become familiar and develop a facility for these diagrammatic computations. The text describes the essential topological ideas through metaphors that are experienced in everyday life: shadows, the human form, the intersections between walls, and the creases in a shirt or a pair of trousers. Mathematically informed reader will benefit from the informal introduction of ideas. This volume will also appeal to scientifically literate individuals who appreciate mathematical beauty.

Diagrammatic Algebra

Diagrammatic Algebra
Author: J. Scott Carter
Publisher: American Mathematical Society
Total Pages: 365
Release: 2021-12-15
Genre: Mathematics
ISBN: 1470466716
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This book is an introduction to techniques and results in diagrammatic algebra. It starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory. The text is replete with figures, diagrams, and suggestive typography that allows the reader a glimpse into many higher dimensional processes. The penultimate chapter summarizes the previous material by demonstrating how to braid 3- and 4- dimensional manifolds into 5- and 6-dimensional spaces. The book is accessible to post-qualifier graduate students, and will also be of interest to algebraists, topologists and algebraic topologists who would like to incorporate diagrammatic techniques into their research.

Hopf Algebras

Hopf Algebras
Author: David E Radford
Publisher: World Scientific
Total Pages: 584
Release: 2011-12-28
Genre: Mathematics
ISBN: 9814405108
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The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph.

Algebraic Invariants of Links

Algebraic Invariants of Links
Author: Jonathan Arthur Hillman
Publisher: World Scientific
Total Pages: 370
Release: 2012
Genre: Mathematics
ISBN: 9814407399
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This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters OCo twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

Algebraic Invariants Of Links (2nd Edition)

Algebraic Invariants Of Links (2nd Edition)
Author: Jonathan Hillman
Publisher: World Scientific
Total Pages: 370
Release: 2012-06-15
Genre: Mathematics
ISBN: 9814407402
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This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology.This second edition introduces two new chapters — twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

Adex Theory: How The Ade Coxeter Graphs Unify Mathematics And Physics

Adex Theory: How The Ade Coxeter Graphs Unify Mathematics And Physics
Author: Saul-paul Sirag
Publisher: World Scientific
Total Pages: 277
Release: 2016-01-25
Genre: Mathematics
ISBN: 9814656518
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This book shows how the ADE Coxeter graphs unify at least 20 different types of mathematical structures. These mathematical structures are of great utility in unified field theory, string theory, and other areas of physics.

Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science-volume 3:the "Golden" Paradigm Of Modern Science: Prerequisite For The "Golden" Revolution In Mathematics,computer Science,and Theoretical Natural Sciences

Mathematics Of Harmony As A New Interdisciplinary Direction And
Author: Alexey Stakhov
Publisher: World Scientific
Total Pages: 244
Release: 2020-09-03
Genre: Mathematics
ISBN: 9811213518
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Volume III is the third part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science - Volume 2: Algorithmic Measurement Theory, Fibonacci And Golden Arithmetic's And Ternary Mirror-symmetrical Arithmetic

Mathematics Of Harmony As A New Interdisciplinary Direction And
Author: Alexey Stakhov
Publisher: World Scientific
Total Pages: 331
Release: 2020-09-03
Genre: Mathematics
ISBN: 9811213488
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Volume II is the second part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

New Ideas In Low Dimensional Topology

New Ideas In Low Dimensional Topology
Author: Vassily Olegovich Manturov
Publisher: World Scientific
Total Pages: 541
Release: 2015-01-27
Genre: Mathematics
ISBN: 9814630632
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This book consists of a selection of articles devoted to new ideas and developments in low dimensional topology. Low dimensions refer to dimensions three and four for the topology of manifolds and their submanifolds. Thus we have papers related to both manifolds and to knotted submanifolds of dimension one in three (classical knot theory) and two in four (surfaces in four dimensional spaces). Some of the work involves virtual knot theory where the knots are abstractions of classical knots but can be represented by knots embedded in surfaces. This leads both to new interactions with classical topology and to new interactions with essential combinatorics.