Braids And Self-distributivity (Progress in Mathematics)

Book Description

This is the award-winning monograph of the Sunyer i Balaguer Prize 1999.

The aim of this book is to present recently discovered connections between Artinâ€™s braid groups and left self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Order properties are crucial.
In the 1980s new examples of left self-distributive systems were discovered using unprovable axioms of set theory, and purely algebraic statements were deduced. The quest for elementary proofs of these statements led to a general theory of self-distributivity centered on a certain group that captures the geometrical properties of this identity. This group happens to be closely connected with Artinâ€™s braid groups, and new properties of the braids naturally arose as an application, in particular the existence of a left invariant linear order, which subsequently received alternative topological constructions.
The text proposes a first synthesis of this area of research. Three domains are considered here, namely braids, self-distributive systems, and set theory. Although not a comprehensive course on these subjects, the exposition is self-contained, and a number of basic results are established. In particular, the first chapters include a rather complete algebraic study of Artinâ€™s braid groups.

The Algebraic Eigenvalue Problem (Numerical Mathematics and Scientific Computation)

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Does Not Get Any Better Than This...

The Algebraic Eigenvalue Problem (Numerical Mathematics and Scientific Computation)
J. Harvie Wilkinson
Manufacturer: Oxford University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0198534183

Book Description

This is a new paperback edition of probably the most important and widely read text on numerical analysis. It is the distillation of what the author learnt about the computation of matrix eigenvalues by dint of trying all methods known to him on computers he helped to make. He was establishing the special nature of these computations as he wrote, and this, along with his emphasis on understanding rather than proving theorems, resulted in an informal and very readable work which became an almost instant classic.

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Does Not Get Any Better Than This..........2007-04-08

This book is really something else.

I found it in my boss' collection, and my appreciation to his taste of books doubled instantly as I started going over the pages. I was aware of the book through Golub and Van Loan's endless citations to it (as AEP) whenever they reach a hard to prove theorem on eigenvalues.

Wilkinson presents hardcore material with clarity, I really appreciate the authorship. For those who share my background (electrical engineering), the authorship is similar to Gallager's "Information Theory and Reliable Communication", or Wozencraft's "Principles of Communication Engineering".

Wilkinson is surprising in this book however. Immediately, after hardcore numerical stability bound derivations, he starts giving practical examples, does not appear to talk down to the reader.

This book is a treasure. Wilkinson is my hero!

Very likely, the book by Parlett "Symmetric Eigenvalue Problem" will be a good companion. When someone borrows your AEP, you can continue learning from SEP.

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Ordered Sets
B. S. W. Schroder
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 0817641289

Book Description

This work is an introduction to the basic tools of the theory of (partially) ordered sets such as visualization via diagrams, subsets, homomorphisms, important order-theoretical constructions, and classes of ordered sets. Using a thematic approach, the author presents open or recently solved problems to motivate the development of constructions and investigations for new classes of ordered sets.

A wide range of material is presented, from classical results such as Dilworth's, Szpilrajn's and Hashimoto's Theorems to more recent results such as the Li--Milner Structure Theorem. Major topics covered include: chains and antichains, lowest upper and greatest lower bounds, retractions, lattices, the dimension of ordered sets, interval orders, lexicographic sums, products, enumeration, algorithmic approaches and the role of algebraic topology.

Since there are few prerequisites, the text can be used as a focused follow-up or companion to a first proof (set theory and relations) or graph theory class. After working through a comparatively lean core, the reader can choose from a diverse range of topics such as structure theory, enumeration or algorithmic aspects. Also presented are some key topics less customary to discrete mathematics/graph theory, including a concise introduction to homology for graphs, and the presentation of forward checking as a more efficient alternative to the standard backtracking algorithm. The coverage throughout provides a solid foundation upon which research can be started by a mathematically mature reader.

Rich in exercises, illustrations, and open problems, Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications.

Lattices and Ordered Algebraic Structures (Universitext)

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Lattices and Ordered Algebraic Structures (Universitext)
T.S. Blyth
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 1852339055

Book Description

Lattices and Ordered Algebraic Structures provides a lucid and concise introduction to the basic results concerning the notion of an order. Although as a whole it is mainly intended for beginning postgraduates, the prerequisities are minimal and selected parts can profitably be used to broaden the horizon of the advanced undergraduate.

The treatment is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. Topics covered include: residuated mappings; Galois connections; modular, distributive, and complemented lattices; Boolean algebras; pseudocomplemented lattices; Stone algebras; Heyting algebras; ordered groups; lattice-ordered groups; representable groups; Archimedean ordered structures; ordered semigroups; naturally ordered regular and inverse Dubreil-Jacotin semigroups.

Featuring material that has been hitherto available only in research articles, and an account of the range of applications of the theory, there are also many illustrative examples and numerous exercises throughout, making it ideal for use as a course text, or as a basic introduction to the field for researchers in mathematics, logic and computer science.

Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture (Chicago Lectures in Mathematics)

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Very interesting mathematics

Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture (Chicago Lectures in Mathematics)
Lionel Schwartz
Manufacturer: University Of Chicago Press
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Binding: Paperback

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ASIN: 0226742032

Book Description

A comprehensive account of one of the main directions of algebraic topology, this book focuses on the Sullivan conjecture and its generalizations and applications. Lionel Schwartz collects here for the first time some of the most innovative work on the theory of modules over the Steenrod algebra, including ideas on the Segal conjecture, work from the late 1970s by Adams and Wilkerson, and topics in algebraic group representation theory.

This course-tested book provides a valuable reference for algebraic topologists and includes foundational material essential for graduate study.

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Very interesting mathematics.......2002-07-21

Fixed points of mappings are of course of enormous interested in the fields of dynamical systems and differential equations. This book however is interested in fixed points from the standpoint of algebraic topology, namely with the study of the homotopy type of the fixed point set of a group action. This motivates the consideration of a homotopy fixed point set, and the author studies specifically the homotopy fixed point set of a finite group acting on a finite complex. From an equivariant point of view, a homotopy fixed point is a set of maps equivariant under the integers modulo 2 (Z/2) from the "antipodal" sphere (i.e the ordinary sphere provided with the antipodal action) to a finite Z/2-CW-complex. A homotopy fixed point is such a map, and ordinary fixed points determine homotopy fixed points. The Sullivan fixed point conjecture asserts that the mapping of ordinary fixed points to homotopy fixed points is a homotopy equivalence, and this conjecture is one of the main topics of the book. The resolution of this conjecture was accomplished by Haynes Miller, with other contributions made by Gunnar Carlsson and J. Lannes. As a point of historical fact, Miller first proved the Sullivan conjecture in the context of pointed maps from the classifying space of a finite group to a finite CW-complex. With the compact-open topology, he showed that this space of maps is weakly contractible. Another line of thought on this topic and considered in this book is that of the fixed point set of a G-space localized at a prime p. The question of whether this fixed point set is weakly homotopy equivalent to the homotopy fixed point set of G acting on the p-localization of a finite CW-complex was solved by Carlsson via a consideration of the Segal conjecture.

The author gives a nice overview of these results, and does so by first considering background material from the theory of unstable modules over the Steenrod algebra. The reader is expected to have a solid background in algebraic topology, particularly in the homotopy theory of CW-complexes, Eilenberg-Maclane spaces, Postnikov systems, the theory of spectral reduced and unreduced cohomology, cohomology operations, and K-theory. The Steenrod algebra has its origins in the consideration of stable Z/2 cohomology operations, where these operations can all be written in terms of Steenrod operations. Consideration of relations among the Steenrod squares result in a family of relations called the Adem relations. This construction can be generalized to a prime p by considering generators other than the Steenrod squares, and dividing out the Adem relations (these are more complex than for the case p = 2). The calculation of the cohomology of Eilenberg-Maclane spaces leads to a characterization of the Steenrod algebra as the algebra of all transformations of mod p cohomology that commute with suspension. Such transformations are called 'stable'.
The mod p cohomology of a space as a module over the Steenrod algebra is unstable, meaning that it is trivial in negative degrees. The author then characterizes the category of unstable modules over the Steenrod algebra (designated U by the author), and shows that is has enough projectives and that it is (locally) Noetherian. That this category has enough injectives is shown using Brown-Gitler technology. This involves the construction of the Brown-Gitler modules, which are related to the Milnor algebra (the dual of the Steenrod algebra, familiar from the elementary theory), and the Carlsson modules. The later are related to Carlsson's work on the Segal conjecture, and their description involves some interesting use of the combinatorics of binary trees. The Lannes functor is introduced as a generalization of this tensor product that still gives an injective category, and its properties are outlined in detail. Modular representation theory is used in the book to study indecomposable reduced U-injectives, and their graded vector space structure is studied using the familiar Poincare series. Then the quotient category of U by its subcategory of nilpotents is studied via a filtration on it, the quotient categories of this filtration being identified with the modular representations of the symmetric groups.
The last part of the book finally gets down to the Sullivan conjecture, beginning with a discussion of the Andre-Quillen cohomology of unstable algebras over the Steenrod algebra. All of the familiar tools from algebraic topology, such as Eilenberg-Moore spectral sequences and the Borel construction are used to prove Miller's version of the Sullivan conjecture and also a generalized version of it.

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Motivic Homotopy Theory (UNIVERSITEXT)
B. Dundas
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Lecture Notes on Motivic Cohomology (Clay Mathematics Monographs) (Clay Mathematics Monographs)

ASIN: 3540458956

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Quantum Invariants: A Study of Knot, 3-Manifolds, and Their Sets
Tomotada Ohtsuki
Manufacturer: World Scientific Publishing Company
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Binding: Hardcover

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ASIN: 9810246757

Book Description

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.

Heights Of Polynomials And Entropy In Algebraic Dynamics (Universitext)

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Heights Of Polynomials And Entropy In Algebraic Dynamics (Universitext)
G. Everest
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 1852331259

Book Description

The main theme of the book is the theory of heights as they appear in various guises. This includes a large body of results on Mahler's measure of the height of a polynomial of which topic there is no book available. The genesis of the measure in a paper by Lehmer is looked at, which is extremely well-timed due to the revival of interest following the work of Boyd and Deninger on special values of Mahler's measure. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations. A large chunk of the book has been devoted to the elliptic Mahler's measure. Special calculation have been included and will be self-contained. One of the most important results about Mahler's measure is that it is the entropy associated to a dynamical system. The authors devote space to discussing this and to giving some convincing and original examples to explain this phenomenon.

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Interesting overview of modern developments in K-theory

Handbook of K-Theory, 2 volume set
E. M., Ed. Friedlander
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 354023019X

Book Description

This handbook offers a compilation of techniques and results in K-theory.

These two volumes consist of chapters, each of which is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. The overall intent of this handbook is to offer the interested reader an exposition of our current state of knowledge as well as an implicit blueprint for future research. This handbook should be especially useful for students wishing to obtain an overview of K-theory and for mathematicians interested in pursuing challenges in this rapidly expanding field.

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Interesting overview of modern developments in K-theory.......2006-01-31

K-theory is now a highly developed but esoteric subject, and touches many different areas in mathematics, including operator theory and functional analysis, algebraic geometry, and geometric topology. In addition, it has found its way into theoretical physics, thanks to the advent of string theory and its more modern metamorphosis M-theory. Everything about K-theory is fascinating, and this two-volume set gives a general overview of the subject from the standpoint of a collection of researchers who have been involved in its development. It is not written for those who are interested in learning K-theory, since it emphasizes developments lying in the frontier of the subject. Students of K-theory, and non-experts (such as this reviewer) can still gain a lot however from its perusal, due to the clarity exhibited in each article along with the copious references at the end of each. Historians of mathematics who want to trace the history of K-theory will also find the volumes of great interest.

K-theory has been developed in both a topological and algebraic context, with the former being more easily grasped for newcomers. It is in the context of algebraic geometry where research in K-theory has shown the greatest activity. Earlier developments in K-theory emphasized its role in the classification and study of vector bundles, and these developments led many to find suitable formulations for algebraic varieties and general schemes. What is now called `motive theory' involves the study of how well known constructions in algebraic topology can be carried over to algebraic geometry. One article in this handbook that gives a good motivation for this study is the one by Daniel Grayson on the motivic spectral sequence. In the article Grayson discusses different approaches to finding a `motivic' version of the Atiyah-Hirzebruch spectral sequence, the latter of which relates topological K-theory to singular cohomology. The trick is not only to find a suitable spectral sequence but also one that is computable. The author shows various ways in which spectral sequences can be constructed, such as the use of long exact sequences in homotopy theory and by using filtrations of a spectrum (such as the familiar Postnikov tower of a space). These are well known in algebraic topology, but for (nonsingular) varieties or (regular) schemes in algebraic geometry one needs another approach that respects as much as possible the general ideas in algebraic topology. One of the approaches discussed is actually fairly intuitive, since it relates K-theory to chain complexes, the latter of which are constructed from direct-sum Grothendieck groups of commuting automorphisms. This approach reflects the well-known strategy of studying the behavior of groups by relating them to the homotopy of a particular space (the mathematician Daniel Quillen used this idea to arrive at his definition of the higher K-groups). Grayson also discusses another approach to obtaining motivic cohomology by using the higher Chow groups, and the work of the mathematician Vladimir Voevodsky on using (affine) homotopy theory of schemes. Voevodsky's work is also motivated by a familiar idea in algebraic topology, namely that of a simplicial space. Voevodsky replaces the simplices by affine spaces over a field, along with the smooth varieties over this field and the colimits of diagrams between these varieties. The colimits are presheaves on these varieties, which are then made into sheaves in a topology called the Nisnevich topology (which is finer than the Zariski topology but coarser than the etale topology). The affine simplices are contractible, and allow the usual techniques of algebraic topology to be applied. In particular, spectra can be defined, called `motivic spectra', and the algebraic K-theory of these spectra results in the motivic spectrum. The Voevodsky construction of a motivic spectral sequence uses a suitable filtration of this motivic spectrum, called the `slice filtration.' The slice filtration involves taking suspensions of the suspension spectra of smooth varieties. Grayson discusses the viability of this approach via the conjectures that were made by Voevodsky, one of which was proved when the field is assumed to have characteristic zero.

Jonathan Rosenberg writes another interesting article in the handbook on the use of K-theory in geometric topology. One immediately thinks of vector bundles in this context and indeed Rosenberg outlines the role of K-theory in the study of flat vector bundles. The K-theory spectrum of the complex numbers arises here, in that every class in this spectrum arises from some flat vector bundle over a homology n-sphere. Also discussed, and definitely a more contemporary topic, is the Waldhausen A-theory, which is a variant of algebraic K-theory, and is highly complex in both its formulation and the proof of its main results. Rosenberg shows how to define the Waldhausen A(X) when X is a pointed space in terms of the infinite loop space whose homotopy groups are the stable homotopy groups of the loop group of X. With multiplication defined by concatenation of loops, this loop space is a `homotopy ring' and if X is path-connected there is map from the loop space to the group ring of the first homotopy group of X. A(X) is then the K-theory space of the ring up to homotopy. The advantage of A(X) according to Rosenberg is that there is essentially a linear map from it to the K-group of the first homotopy group ring, which in some cases is an equivalence. For the case of compact smooth manifolds and its space of pseudo-isotopies, A(X) gives information on higher Whitehead and Reidemeister torsion. Rosenberg ends the article with a very brief discussion of the application of K-theory to symbolic dynamics. In this area of chaotic dynamical systems one is interested in what transition matrices will give equivalent symbolic dynamics. One can define an equivalence relation between them, called `shift equivalence.' K-theory assists in the study of shift equivalence by defining a C*-algebra associated to the shift, and studying the zeroth K-group of this C*-algebra.

The Brauer-Hasse-Noether Theorem in Historical Perspective (Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften)

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The Brauer-Hasse-Noether Theorem in Historical Perspective (Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften)
P. Roquette
Manufacturer: Springer
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ASIN: 354023005X

Book Description

The unpublished writings of Helmut Hasse, consisting of letters, manuscripts and other papers, are kept at the Handschriftenabteilung of the University Library at GÃ¶ttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, results and methods freely with his colleagues. There are more than 8000 documents preserved. Although not all of them are of equal mathematical interest, searching through this treasure can help us to assess the development of Number Theory through the 1920s and 1930s.

The present volume is largely based on the letters and other documents its author has found concerning the Brauer-Hasse-Noether Theorem in the theory of algebras; this covers the years around 1931. In addition to the documents from the literary estates of Hasse and Brauer in GÃ¶ttingen, the author also makes use of some letters from Emmy Noether to Richard Brauer that are preserved at the Bryn Mawr College Library (Pennsylvania, USA).

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