Part of London Mathematical Society Student Texts
Publication planned for: July 2018
availability: Not yet published - available from July 2018
format: Hardback
isbn: 9781108425919
format: Paperback
isbn: 9781108441834
This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.
Introduction
1. Algebras and modules
2. Functors between module categories
3. Character theory
4. Algebras over p-local rings
5. Group algebras and modules over p-local rings
References
Index.
Part of London Mathematical Society Lecture Note Series
Publication planned for: May 2018
availability: Not yet published - available from April 2018
format: Paperback
isbn: 9781108447232
This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.
Introduction
Part I. Toposes and Differential Geometry:
1. Topos theory
2. Synthetic differential geometry
Part II. Topics in SDG:
3. The Ambrose-Palais-Singer theorem in SDG
4. Calculus of variations in SDG
Part III. Toposes and Differential Topology:
5. Local concepts in SDG
6. Synthetic differential topology
Part IV. Topics in SDT:
7. Stable mappings and Mather's theorem in SDT
8. Morse theory in SDT
Part V. SDT and Differential Topology:
9. Well-adapted models of SDT
10. An application to unfoldings
Part VI. A Well-Adapted Model of SDT:
11. The Dubuc topos G
12. G as a model of SDT
References
Index.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: August 2018
availability: Not yet published - available from August 2018
format: Hardback
isbn: 9781107189928
This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects and extends many results scattered throughout the literature, in particular the Langlands constant term formula for Eisenstein series on G(A) as well as the Casselman-Shalika formula for the p-adic spherical Whittaker function. This book also covers more advanced topics such as spherical Hecke algebras and automorphic L-functions. Many of these mathematical results have natural interpretations in string theory, and so some basic concepts of string theory are introduced with an emphasis on connections with automorphic forms. Throughout the book special attention is paid to small automorphic representations, which are of particular importance in string theory but are also of independent mathematical interest. Numerous open questions, and conjectures motivated by physics, are included to prompt the reader's own research.
1. Motivation and background
Part I. Automorphic Representations:
2. Preliminaries on p-adic and adelic technology
3. Basic notions from Lie algebras and Lie groups
4. Automorphic forms
5. Automorphic representations and Eisenstein series
6. Whittaker functions and Fourier coefficients
7. Fourier coefficients of Eisenstein series on SL(2, A)
8. Langlands constant term formula
9. Whittaker coefficients of Eisenstein series
10. Analysing Eisenstein series and small representations
11. Hecke theory and automorphic L-functions
12. Theta correspondences
Part II. Applications in String Theory:
13. Elements of string theory
14. Automorphic scattering amplitudes
15. Further occurrences of automorphic forms in string theory
Part III. Advanced Topics:
16. Connections to the Langlands program
17. Whittaker functions, crystals and multiple Dirichlet series
18. Automorphic forms on non-split real forms
19. Extension to Kac?Moody groups
Appendix A. SL(2, R) Eisenstein series and Poisson resummation
Appendix B. Laplace operators on G/K and automorphic forms
Appendix C. Structure theory of su(2, 1)
Appendix D. Poincare series and Kloosterman sums
References
Index.
Hardback
January 30, 2018 Forthcoming by Chapman and Hall/CRC
Textbook - 578 Pages - 28 B/W Illustrations
ISBN 9781138063471 - CAT# K33364
Series: Textbooks in Mathematics
Over 800 exercises, projects, and computer explorations
Increased coverage of cryptography, including Vigenere, Stream, Transposition,
and Block ciphers, along with RSA and discrete log-based systems
"Check Your Understanding" questions for instant feedback to students
New Appendices on "What is a proof?" and on Matrices
Select basic (pre-RSA) cryptography now placed in an earlier chapter so that the topic can be covered right after the basic material on congruences
Answers and hints for odd-numbered problems
Building on the success of the first edition, An Introduction to Number Theory with Cryptography, Second Edition, increases coverage of the popular and important topic of cryptography, integrating it with traditional topics in number theory.
The authors have written the text in an engaging style to reflect number theory's increasing popularity. The book is designed to be used by sophomore, junior, and senior undergraduates, but it is also accessible to advanced high school students and is appropriate for independent study. It includes a few more advanced topics for students who wish to explore beyond the traditional curriculum.
20 1. Introduction; 2 Divisibility; 3. Linear Diophantine Equations; 4. Unique Factorization; 5. Applications of Unique Factorization; 6. Conguences; 7. Classsical Cryposystems; 8. Fermat, Euler, Wilson; 9. RSA; 10. Polynomial Congruences; 11. Order and Primitive Roots; 12. More Cryptographic Applications; 13. Quadratic Reciprocity; 14. Primality and Factorization; 15. Geometry of Numbers; 16. Arithmetic Functions; 17. Continued Fractions; 18. Gaussian Integers; 19. Algebraic Integers; 20. Analytic Methods, 21. Epilogue: Fermat's Last Theorem; Appendices; Answers and Hints for Odd-Numbered Exercises; Index
Hardback
March 13, 2018 Forthcoming by CRC Press
Reference - 528 Pages - 289 B/W Illustrations
ISBN 9781138561243 - CAT# K43463
Describes the main concepts of modern knot theory from its classical roots to the latest developments
Includes Khovanov's categorification construction for the Jones polynomial, with all proofs
Explores braid theory, encoding by d-diagrams, and virtual knot theory
Addresses Vassiliev's knot invariants as well as Bar-Natan's theory on Lie algebra representation and knots
Introduces Legendrian knot theory together with Chekanov's DGA construction
Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. It suffices to mention the great progress in knot homology theory (Khovanov homology and Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give rise to strong invariants of knots and 3-manifolds, in particular, many new unknot detectors. New to the Second Edition; discusses Heegaard-Floer homology theory and A-polynomial of classical links. Updates throughout the book.
Knots, links, and invariant polynomials. Introduction. Reidemeister moves. Knot arithmetics. Links in 2-surfaces in R3.Fundamental group; the knot group. The knot quandle and the Conway algebra. Kauffman's approach to Jones polynomial. Properties of Jones polynomials. Khovanov's complex. Theory of braids. Braids, links and representations of braid groups. Braids and links. Braid construction algorithms. Algorithms of braid recognition. Markov's theorem; the Yang-Baxter equation. Vassiliev's invariants. Definition and Basic notions of Vassiliev invariant theory. The chord diagram algebra. The Kontsevich integral and formulae for the Vassiliev invariants. Atoms and d-diagrams. Atoms, height atoms and knots. The bracket semigroup of knots. Virtual knots. Basic definitions and motivation. Invariant polynomials of virtual links. Generalised Jones-Kauffman polynomial. Long virtual knots and their invariants. Virtual braids. Other theories. 3-manifolds and knots in 3-manifolds. Legendrian knots and their invariants. Independence of Reidemeister moves.