EDITED BY
S. MONTGOMERY AND HANS-JURGEN SCHNEIDER

New Directions in Hopf Algebras

Description: This timely collection of expository papers by leading researchers in the field highlights progress and new directions in Hopf algebras. The volume arises from the MSRI workshop on Hopf Algebras in October 1999, where many exciting recent developments were discussed. The work presented here will be stimulating to researchers and accessible to graduate students. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to recent work in quantum groups. In particular, there are chapters on recent progress in classifying finite-dimensional Hopf algebras, both in the semisimple case and in the pointed case, as well as what is known about the extension theory of Hopf algebras and on important connections of Hopf algebras to Lie algebras, knot theory and operator algebras. The volume also includes Mitsuhiro Takeuchi’s article ‘A short course on quantum matrices’, now a standard reference in spite of its relative lack of availability; it has been updated for this volume.

Contents: 1. Pointed Hopf algebras Nicolas Andruskiewitsch and Hans-Jurgen Schneider; 2. On the classification of finite-dimensional triangular Hopf algebras Shlomo Gelaki; 3. Coideal subalgebras and quantum symmetric pairs Gail Letzter; 4. Hopf algebra extensions and cohomology Akira Masuoka; 5. Finite quantum groupoids and their applications Dmitri Nikshych and Leonid Vainerman; 6. On quantum algebras and coalgebras, oriented quantum algebras and coalgebras, invariants of 1-1 tangles, knots, and links David Radford; 7. Hopf algebra extensions and monoidal categories Peter Schauenburg; 8. A short course on quantum matrices Mitsuhiro Takeuchi; 9. The Brauer group of a Hopf algebra Fred Van Oystaeyen and Yinhuo Zhang.

Essential Information
ISBN, Binding, : 0-521-81512-6 Hardback
Pages: 496
Approximate Publication Date: c.01/07/2002
Main Subject Category: Algebra
Series: Mathematical Sciences Research Institute Publications, No. 43

Contributors: Nicolas Andruskiewitsch, Hans-Jurgen Schneider, Shlomo Gelaki, Gail Letzter, Akira Masuoka, Dmitri Nikshych, Leonid Vainerman, David Radford, Peter Schauenburg, Mitsuhiro Takeuchi, Fred Van Oystaeyen, Yinhuo Zhang

Market (Subject)
algebra, geometry, mathematical physics

Level
graduate students, academic researchers

Comparable titles: BILLERA et al./New Perspectives in Algebraic Combinatorics/1999/0521 770874

GEORGE TOURLAKIS

Logic and Set Theory

Volume 1, Mathematical Logic

Description: This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume I includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel’s 2nd incompleteness since Hilbert and Bernay’s Grundlagen.

Contents: 1. Basic logic; 2. The second incompleteness theorem.

Essential Information
ISBN, Binding, : 0-521-75373-2 Hardback
Pages: 300
Approximate Publication Date: c.01/07/2002
Main Subject Category: Maths - foundations, combinatorics
Series: Cambridge Studies in Advanced Mathematics

Market (Subject)
mathematical logic, philosophical logic, theoretical computer science

Level
graduate students, academic researchers, undergraduate students

Comparable titles: MACHOVER/Set Theory, Logic and Limitations/1996/0521 474930

VERN PAULSEN
University of Houston

Completely Bounded Maps and Operator Algebras

Description: In this book the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensable introduction to the theory of operator spaces for all who want to know more.

Contents: 1. Introduction; 2. Positive maps; 3. Completely positive maps; 4. Dilation theorems; 5. Commuting contractions; 6. Completely positive maps into Mn; 7. Arveson’s extension theorems; 8. Completely bounded maps; 9. Completely bounded homomorphisms; 10. Polynomially bounded operators; 11. Applications to K-spectral sets; 12. Tensor products and joint spectral sets; 13. Operator systems and operator spaces; 14. An operator space bestiary; 15. Injective envelopes; 16. Multipliers and operator algebras; 17. Completely bounded multilinear maps; 18. Applications of operator algebras; 19. Further results on injectivity.

Essential Information
ISBN, Binding, : 0-521-81669-6 Hardback
Pages: 320
Approximate Publication Date: c.01/07/2002
Main Subject Category: Mathematics - analysis, probability
Series: Cambridge Studies in Advanced Mathematics, No. 78

Market (Subject)
mathematics

Level
graduate students, academic researchers

IOAN JAMES
University of Oxford

Remarkable Mathematicians

Description: Ioan James introduces and profiles sixty mathematicians from an era which saw mathematics freed from its classical origins to develop into its modern form. The characters, all born between 1700 and 1910, come from a wide range of countries, and all made an important contribution to mathematics, through their ideas, their teaching, their influence, and so on. The book is organised chronologically into ten chapters, each of which contains potted life stories of six mathematicians. The players James has chosen to portray are sufficiently representative that their stories, when read in sequence, convey in human terms something of the way in which mathematics developed.

Contents: Preface; 1. From Euler to Legendre; 2. From Fourier to Cauchy; 3. From Abel to Grassmann; 4. From Kummer to Cayley; 5. From Hermite to Lie; 6. From Cantor to Hilbert; 7. From Moore to Takagi; 8. From Hardy to Lefschetz; 9. From Birkhoff to Alexander; 10. From Banach to von Neumann; Epilogue; Further reading.

Essential Information
First Author: James
Title: Remarkable Mathematicians
ISBN, Binding, : 0-521-52094-0 Paperback
Pages: 320
Approximate Publication Date: c.01/10/2002
Main Subject Category: Mathematics (general)

Market (Subject)
mathematics

Level
graduate students, academic researchers, general readers, undergraduate students

Bibliographic Details
60 figures

Comparable titles: BOLLOBAS/Littlewood's Miscellany/1986/0521 33702X

PATRICK BLACKBURN
LORIA, Nancy

Modal Logic

Description: Now available in paperback, this is a modern, advanced textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realised its worth. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. Ideal for anyone wanting to learn modern modal logic.

Contents: 1. Basic concepts; 2. Models; 3. Frames; 4. Completeness; 5. Algebras and general frames; 6. Computability and complexity; 7. Extended modal logic.

Essential Information
ISBN, Binding, : 0-521-52714-7 Paperback
Pages: 554
Approximate Publication Date: c.01/07/2002
Main Subject Category: Theory of computation, data
Series: Cambridge Tracts in Theoretical Computer Science, No. 53

Market (Subject)
computer science, logic

Level
graduate students, academic researchers

Bibliographic Details
20 line diagrams

Comparable titles: BLACKBURN et al./Modal Logic/2001/0521 802008