Download or read book An Introduction to Quantum and Vassiliev Knot Invariants written by David M. Jackson and published by Springer. This book was released on 2019-05-04 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.
Download or read book Equivalence Invariants and Symmetry written by Peter J. Olver and published by Cambridge University Press. This book was released on 1995-06-30 with total page 546 pages. Available in PDF, EPUB and Kindle. Book excerpt: Drawing on a wide range of mathematical disciplines, including geometry, analysis, applied mathematics and algebra, this book presents an innovative synthesis of methods used to study problems of equivalence and symmetry which arise in a variety of mathematical fields and physical applications. Systematic and constructive methods for solving equivalence problems and calculating symmetries are developed and applied to a wide variety of mathematical systems, including differential equations, variational problems, manifolds, Riemannian metrics, polynomials and differential operators. Particular emphasis is given to the construction and classification of invariants, and to the reductions of complicated objects to simple canonical forms. This book will be a valuable resource for students and researchers in geometry, analysis, algebra, mathematical physics and other related fields.
Download or read book Symmetry Representations and Invariants written by Roe Goodman and published by Springer Science & Business Media. This book was released on 2009-07-30 with total page 731 pages. Available in PDF, EPUB and Kindle. Book excerpt: Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.
Download or read book Invariants of Homology 3 Spheres written by Nikolai Saveliev and published by Springer Science & Business Media. This book was released on 2002-09-05 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives a systematic exposition of the diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered include: constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, and Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. The text will be a valuable source for both the graduate student and researcher in mathematics and theoretical physics.
Download or read book Quantum Invariants written by Tomotada Ohtsuki and published by World Scientific. This book was released on 2002 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern-Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants.
Download or read book Quantum Invariants written by Tomotada Ohtsuki and published by World Scientific. This book was released on 2001-12-21 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants. Contents: Knots and Polynomial InvariantsBraids and Representations of the Braid GroupsOperator Invariants of Tangles via Sliced DiagramsRibbon Hopf Algebras and Invariants of LinksMonodromy Representations of the Braid Groups Derived from the Knizhnik–Zamolodchikov EquationThe Kontsevich InvariantVassiliev InvariantsQuantum Invariants of 3-ManifoldsPerturbative Invariants of Knots and 3-ManifoldsThe LMO InvariantFinite Type Invariants of Integral Homology 3-Spheres Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics. Keywords:Kontsevich Invariant;LMO Invariant;Quantum Groups;Knot;3-Manifold;Quantum Invariant;Vassiliev Invariant;Finite Type Invariant;Chord Diagram;Jacobi Diagram;KZ Equation;Chern-Simons TheoryReviews:“This is a nicely written and useful book: I think that the author has done a great job in explaining quantum invariants of knots and 3-manifolds also on an intuitive and well-motivated, organically growing and not too technical level, at the same time however presenting a lot of material ordered by a clear guiding line.”Mathematics Abstracts “Ohtsuki's book is a very valuable addition to the literature. It surveys the full spectrum of work in the area of quantum invariants … Ohtsuk's book is very readable, for he makes an attempt to present the material in as straightforward a way as possible … the presentation here is very clear and should be easily accessible … this is an excellent book which I would recommend to beginners wanting to learn about quantum invariants and to experts alike.”Mathematical Reviews
Download or read book Introduction to l2 invariants written by Holger Kammeyer and published by Springer Nature. This book was released on 2019-10-29 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the reader to the most important concepts and problems in the field of l2-invariants. After some foundational material on group von Neumann algebras, l2-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of l2-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of l2-Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lück's approximation theorem and its generalizations. The final chapter deals with l2-torsion, twisted variants and the conjectures relating them to torsion growth in homology. The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.
Download or read book Moments and Moment Invariants in Pattern Recognition written by Jan Flusser and published by John Wiley & Sons. This book was released on 2009-11-04 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Moments as projections of an image’s intensity onto a proper polynomial basis can be applied to many different aspects of image processing. These include invariant pattern recognition, image normalization, image registration, focus/ defocus measurement, and watermarking. This book presents a survey of both recent and traditional image analysis and pattern recognition methods, based on image moments, and offers new concepts of invariants to linear filtering and implicit invariants. In addition to the theory, attention is paid to efficient algorithms for moment computation in a discrete domain, and to computational aspects of orthogonal moments. The authors also illustrate the theory through practical examples, demonstrating moment invariants in real applications across computer vision, remote sensing and medical imaging. Key features: Presents a systematic review of the basic definitions and properties of moments covering geometric moments and complex moments. Considers invariants to traditional transforms – translation, rotation, scaling, and affine transform - from a new point of view, which offers new possibilities of designing optimal sets of invariants. Reviews and extends a recent field of invariants with respect to convolution/blurring. Introduces implicit moment invariants as a tool for recognizing elastically deformed objects. Compares various classes of orthogonal moments (Legendre, Zernike, Fourier-Mellin, Chebyshev, among others) and demonstrates their application to image reconstruction from moments. Offers comprehensive advice on the construction of various invariants illustrated with practical examples. Includes an accompanying website providing efficient numerical algorithms for moment computation and for constructing invariants of various kinds, with about 250 slides suitable for a graduate university course. Moments and Moment Invariants in Pattern Recognition is ideal for researchers and engineers involved in pattern recognition in medical imaging, remote sensing, robotics and computer vision. Post graduate students in image processing and pattern recognition will also find the book of interest.
Download or read book The Invariants of Linear Differential Expressions written by Frank Irwin and published by . This book was released on 1908 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Invariants of Quadratic Differential Forms written by Oswald Veblen and published by . This book was released on 1927 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: An early tract for students of differential geometry and mathematical physics.
Download or read book Cohomological Invariants in Galois Cohomology written by Skip Garibaldi and published by American Mathematical Soc.. This book was released on 2003 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume addresses algebraic invariants that occur in the confluence of several important areas of mathematics, including number theory, algebra, and arithmetic algebraic geometry. The invariants are analogues for Galois cohomology of the characteristic classes of topology, which have been extremely useful tools in both topology and geometry. It is hoped that these new invariants will prove similarly useful. Early versions of the invariants arose in the attempt to classify the quadratic forms over a given field. The authors are well-known experts in the field. Serre, in particular, is recognized as both a superb mathematician and a master author. His book on Galois cohomology from the 1960s was fundamental to the development of the theory. Merkurjev, also an expert mathematician and author, co-wrote The Book of Involutions (Volume 44 in the AMS Colloquium Publications series), an important work that contains preliminary descriptions of some of the main results on invariants described here. The book also includes letters between Serre and some of the principal developers of the theory. It will be of interest to graduate students and research mathematicians interested in number th
Download or read book Basic Global Relative Invariants for Nonlinear Differential Equations written by Roger Chalkley and published by American Mathematical Soc.. This book was released on 2007 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations. With respect to any fixed integer $\, m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\, \mathcal{C {m,2 $ that contains equations like $Q {m = 0$ in which $Q {m $ is a quadratic form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){2 $ is $1$.Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\, \mathcal{C {m, n $ that contains equations like $H {m, n = 0$ in which $H {m, n $ is an $n$th-degree form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){n $ is $1$.These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equa
Download or read book L2 Invariants Theory and Applications to Geometry and K Theory written by Wolfgang Lück and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt: In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.
Download or read book Derivatives of Links Milnor s Concordance Invariants and Massey s Products written by Tim D. Cochran and published by American Mathematical Soc.. This book was released on 1990 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: We investigate higher-order cohomology operations (Massey products) on complements of links of circles in [italic]S3. These are known to be essentially equivalent to the [lowercase Greek]Mu [with macron]-invariants of John Milnor, which detect whether or not the longitudes of the link lie in the [italic]n[superscript]th term of the lower central series of the fundamental group of the link compliment. We define a geometric "derivative" on the set of all links and use this to define higher-order linking numbers which are shown to be "pieces" of Massey products.
Download or read book Introduction to the Algebraic Theory of Invariants of Differential Equations written by Konstantin Sergeevich Sibirskiĭ and published by Manchester University Press. This book was released on 1988 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considers polynominal invariants & comitants of autonomous systems of differential equations with right-hand sides relative to various transformation groups of phase space. Contains an in-depth discussion of the two-dimensional system with quadratic right-hand sides. Features numerous applications to the qualitative theory of differential equations.
Download or read book Polynomial Invariants of Finite Groups written by Larry Smith and published by CRC Press. This book was released on 1995-04-15 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written by an algebraic topologist motivated by his own desire to learn, this well-written book represents the compilation of the most essential and interesting results and methods in the theory of polynomial invariants of finite groups. From the table of contents: - Invariants and Relative Invariants - Finite Generation of Invariants - Constructio
Download or read book Invariants of Quadratic Differential Forms written by Joseph Edmund Wright and published by Courier Corporation. This book was released on 2013-06-19 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This classic monograph by a mathematician affiliated with Trinity College, Cambridge, offers a brief account of the invariant theory connected with a single quadratic differential form. A historical overview is followed by considerations of the methods of Christoffel and Lie as well as Maschke's symbolic method and explorations of geometrical and dynamical methods. 1960 edition"--
