Series: Australian Mathematical Society Lecture Series (No. 19)
Paperback (ISBN-13: 9780521695244 | ISBN-10: 0521695244)
Algebra has moved well beyond the topics discussed in standard
undergraduate texts on 'modern algebra'. Those books typically
dealt with algebraic structures such as groups, rings and fields:
still very important concepts! However Quantum Groups: A Path to
Current Algebra is written for the reader at ease with at least
one such structure and keen to learn the latest algebraic
concepts and techniques. A key to understanding these new
developments is categorical duality. A quantum group is a vector
space with structure. Part of the structure is standard: a
multiplication making it an 'algebra'. Another part is not in
those standard books at all: a comultiplication, which is dual to
multiplication in the precise sense of category theory, making it
a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras
have been around for half a century, the term 'quantum group',
along with revolutionary new examples, was launched by Drinfel'd
in 1986.
* Essential for any graduate student or researcher whose research
involves quantum groups
* Containing the latest algebraic concepts and techniques, this
book updates the meaning of 'modern algebra'
* Includes over 60 worked examples and exercises
Contents
Introduction; 1. Revision of basic structures; 2. Duality between
geometry and algebra; 3. The quantum general linear group; 4.
Modules and tensor products; 5. Cauchy modules; 6. Algebras; 7.
Coalgebras and bialgebras; 8. Dual coalgebras of algebras; 9.
Hopf algebras; 10. Representations of quantum groups; 11. Tensor
categories; 12. Internal homs and duals; 13. Tensor functors and
Yang-Baxter operators; 14. A tortile Yang-Baxter operator for
each finite-dimensional vector space; 15. Monoids in tensor
categories; 16. Tannaka duality; 17. Adjoining an antipode to a
bialgebra; 18. The quantum general linear group again; 19.
Solutions to exercises; References; Index.
Series: London Mathematical Society Lecture Note Series (No.
338)
Paperback (ISBN-13: 9780521691826 | ISBN-10: 0521691826)
The focus of this book is the continuing strength of pure
mathematics in Russia after the post-Soviet diaspora. The authors
are eight young specialists who have chosen to remain in their
country and who are associated with strong research groups in
Moscow and St. Petersburg in the fields of algebraic geometry and
number theory. Their articles are based on lecture courses given
at British universities. The articles are mainly surveys of the
recent work of the research groups and contain a substantial
number of original results. Topics covered are homogeneous spaces
(embeddings, dual varieties), formal groups, mirror duality, del
Pezzo fibrations, Diophantine approximation and geometric
quantization. The authors are I. Arzhantsev, M. Bondarko, V.
Golyshev, M. Grinenko, N. Moshchevitin, E. Tevelev, D. Timashev
and N. Tyurin. Mathematical researchers and graduate students in
algebraic geometry and number theory worldwide will find this
book of great interest.
* Showcases the continuing vitality of mathematics in Russia
* Presents the research of major groups working in algebraic
geometry and number theory in Russia
* Contains a substantial number of original results
Contents
Preface; 1. Affine embeddings of homogeneous spaces I. V.
Arzhantsev; 2. Formal groups over local fields: a constructive
approach M. V. Bondarko; 3. Classification problems and mirror
duality V. V. Golyshev; 4. Birational models of del Pezzo
fibrations M. M. Grinenko; 5. Best Diophantine approximations N.
G. Moshchevitin; 6. Projectively dual varieties of homogeneous
spaces E. A. Tevelev; 7. Equivariant embeddings of homogeneous
spaces D. A. Timashev; 8. Geometric quantization and algebraic
Lagrangian geometry N. A. Tyurin.
Contributors
I. V. Arzhantsev, M. V. Bondarko, V. V. Golyshev, M. M. Grinenko,
N. G. Moshchevitin, E. A. Tevelev, D. A. Timashev, N. A. Tyurin
Hardback (ISBN-13: 9780521873307 | ISBN-10: 0521873304)
Distributed systems are fast becoming the norm in computer
science. Formal mathematical models and theories of distributed
behaviour are needed in order to understand them. This book
proposes a distributed pi-calculus called Dpi, for describing the
behaviour of mobile agents in a distributed world. It is based on
an existing formal language, the pi-calculus, to which it adds a
network layer and a primitive migration construct. A mathematical
theory of the behaviour of these distributed systems is
developed, in which the presence of types plays a major role. It
is also shown how in principle this theory can be used to develop
verification techniques for guaranteeing the behavior of
distributed agents. The text is accessible to computer scientists
with a minimal background in discrete mathematics. It contains an
elementary account of the pi-calculus, and the associated theory
of bisimulations. It also develops the type theory required by
Dpi from first principles.
* First book on formal foundations of distributed computation
* Accessible introduction to the theory of the pi-calculus, with
many exercises
* Contains many worked examples and over 70 exercises
Contents
1. Introduction; 2. Inductive principles; 3. The asynchronous Pi-Calculus;
4. Types for API; 5. Types and behaviour in API; 6. A distributed
asynchronous Pi-Calculus; Behavioural Equivalences for ADPI;
Sources; List of figures; Notation.
Series: London Mathematical Society Student Texts (No. 68)
Paperback (ISBN-13: 9780521682244 | ISBN-10: 052168224X)
Hardback (ISBN-13: 9780521863605 | ISBN-10: 0521863600)
Written for graduate students, this book presents topics in 2-dimensional
hyperbolic geometry. The authors begin with rigid motions in the
plane which are used as motivation for a full development of
hyperbolic geometry in the unit disk. The approach is to define
metrics from an infinitesimal point of view; first the density is
defined and then the metric via integration. The study of
hyperbolic geometry in arbitrary domains requires the concepts of
surfaces and covering spaces as well as uniformization and
Fuchsian groups. These ideas are developed in the context of what
is used later. The authors then provide a detailed discussion of
hyperbolic geometry for arbitrary plane domains. New material on
hyperbolic and hyperbolic-like metrics is presented. These are
generalizations of the Kobayashi and Caratheodory metrics for
plane domains. The book concludes with applications to
holomorphic dynamics including new results and accessible open
problems.
* Entirely self-contained text on hyperbolic geometry for plane
domains, accessible to upper-level undergraduate and graduate *tudents
yet also suitable for academic researchers
* Contains over 250 exercises and worked examples
* Presents brand new material on hyperbolic and hyperbolic-like
matrices
Contents
Introduction; 1. Elementary transformations; 2 Hyperbolic metric
in the unit disk; 3. Holomorphic functions; 4. Topology and
uniformization; 5. Discontinuous groups; 6 Fuchsian groups; 7.
General hyperbolic metric; 8. The Kobayashi metric; 9. The
Caratheodory pseudo metric; 10. Contraction properties; 11.
Applications; 12 Applications II; 13. Applications III; 14.
Estimating hyperbolic densities; 15. Uniformly perfect domains;
16 Appendix: Elliptic functions; Bibliography.