CRM Monograph Series, Volume: 27
2008; 147 pp; hardcover
ISBN-13: 978-0-8218-4480-9
Expected publication date is January 8, 2009.
The two parts of this text are based on two series of lectures delivered by Jean Berstel and Christophe Reutenauer in March 2007 at the Centre de Recherches Mathematiques, Montreal, Canada. Part I represents the first modern and comprehensive exposition of the theory of Christoffel words. Part II presents numerous combinatorial and algorithmic aspects of repetition-free words stemming from the work of Axel Thue--a pioneer in the theory of combinatorics on words.
A beginner to the theory of combinatorics on words will be motivated by the numerous examples, and the large variety of exercises, which make the book unique at this level of exposition. The clean and streamlined exposition and the extensive bibliography will also be appreciated. After reading this book, beginners should be ready to read modern research papers in this rapidly growing field and contribute their own research to its development.
Experienced readers will be interested in the finitary approach to Sturmian words that Christoffel words offer, as well as the novel geometric and algebraic approach chosen for their exposition. They will also appreciate the historical presentation of the Thue-Morse word and its applications, and the novel results on Abelian repetition-free words.
Graduate students and research mathematicians interested in combinatorics on words, theory of computation, symbolic dynamics, Markoff numbers, continued fractions, group theory, pattern recognition, and stringology.
University Lecture Series, Volume: 48
2008; 147 pp; softcover
ISBN-13: 978-0-8218-4735-0
Expected publication date is January 16, 2009.
Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems. As a model for polyhedral reduction theories of positive definite quadratic forms, Minkowski's classical theory is presented, including an application to multidimensional continued fraction expansions. The reduction theories of Voronoi are described in great detail, including full proofs, new views, and generalizations that cannot be found elsewhere. Based on Voronoi's second reduction theory, the local analysis of sphere coverings and several of its applications are presented. These include the classification of totally real thin number fields, connections to the Minkowski conjecture, and the discovery of new, sometimes surprising, properties of exceptional structures such as the Leech lattice or the root lattices.
Throughout this book, special attention is paid to algorithms and computability, allowing computer-assisted treatments. Although dealing with relatively classical topics that have been worked on extensively by numerous authors, this book is exemplary in showing how computers may help to gain new insights.
Graduate students and research mathematicians interested in the geometry of numbers, discrete geometry, and computational mathematics.
From quadratic forms to sphere packings and coverings
Minkowski reduction
Voronoi I
Voronoi II
Local analysis of coverings and applications
Polyhedral representation conversion under symmetries
Possible future projects
Bibliography
Index
Notations
Mathematical Surveys and Monographs,Volume: 152
2009; 235 pp; hardcover
ISBN-13: 978-0-8218-4691-9
Expected publication date is January 23, 2009.
Based on a lecture series given by the authors at a satellite meeting of the 2006 International Congress of Mathematicians and on many articles written by them and their collaborators, this volume provides a comprehensive up-to-date survey of several core areas of combinatorial geometry. It describes the beginnings of the subject, going back to the nineteenth century (if not to Euclid), and explains why counting incidences and estimating the combinatorial complexity of various arrangements of geometric objects became the theoretical backbone of computational geometry in the 1980s and 1990s. The combinatorial techniques outlined in this book have found applications in many areas of computer science from graph drawing through hidden surface removal and motion planning to frequency allocation in cellular networks.
Combinatorial Geometry and Its Algorithmic Applications is intended as a source book for professional mathematicians and computer scientists as well as for graduate students interested in combinatorics and geometry. Most chapters start with an attractive, simply formulated, but often difficult and only partially answered mathematical question, and describes the most efficient techniques developed for its solution. The text includes many challenging open problems, figures, and an extensive bibliography.
Graduate students and research mathematicians interested in combinatorial geometry and algorithmic applications.
Contemporary Mathematics, Volume: 477
2009; 141 pp; softcover
ISBN-13: 978-0-8218-3984-3
Expected publication date is February 6, 2009.
This volume contains articles representing the courses given at the 2005 RSME Santalo Summer School on "Recent Trends in Cryptography". The main goal of the Summer School was to present some of the recent mathematical methods used in cryptography and cryptanalysis. The School was oriented to graduate and doctoral students, as well as recent doctorates. The material is presented in an expository manner with many examples and references.
The topics in this volume cover some of the most interesting new developments in public key and symmetric key cryptography, such as pairing based cryptography and lattice based cryptanalysis.
Graduate students and research mathematicians interested in cryptography and cryptanalysis.
A. Fuster-Sabater -- Cellular automata in stream ciphers
T. Helleseth -- Linear and nonlinear sequences and applications to stream ciphers
A. Menezes -- An introduction to pairing-based cryptography
P. Q. Nguyen -- Public-key cryptanalysis
I. E. Shparlinski -- Pseudorandom Number Generators from Elliptic Curves
Contemporary Mathematics, Volume: 478
2009; 295 pp; softcover
ISBN-13: 978-0-8218-4555-4
Expected publication date is February 15, 2009.
Articles in this volume cover topics related to representation theory of various algebraic objects such as algebraic groups, quantum groups, Lie algebras, (finite- and infinite-dimensional) finite groups, and quivers. Collected in one book, these articles show deep relations between all these aspects of Representation Theory, as well as the diversity of algebraic, geometric, topological, and categorical techniques used in studying representations.
Graduate students and research mathematicians interested in representation theory.
H. H. Andersen -- Sum formulas and Ext-groups
S. Doty -- Schur-Weyl duality in positive characteristic
A. Francis and W. Wang -- The centers of Iwahori-Hecke algebras are filtered
University of Georgia Vigre Algebra Group, On Kostant's theorem for Lie algebra cohomology
X. He -- G-stable pieces and partial flag varieties
L. Ji -- Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups
S. Kumar, G. Lusztig, and D. Prasad -- Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups
G. Liu -- Classification of finite-dimensional basic Hopf algebras according to their representation type
G. Lusztig -- Twelve bridges from a reductive group to its Langlands dual
B. J. Parshall and L. L. Scott -- Some new highest weight categories
I. Pop and A. Stolin -- Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation
C. M. Ringel -- The relevance and the ubiquity of Prufer modules
A. Savage -- Quivers and the Euclidean group
S. Shang and Y. Gao -- mathfrak {eu}_2-Lie admissible algebras and Steinberg
unitary Lie algebras
T. Shoji -- Lusztig's conjecture for finite classical groups with even characteristic
Y. Su -- A survey on quasifinite representations of Weyl type Lie algebras
N. Xi -- Maximal and primitive elements in baby Verma modules for type B_2
Y.-F. Yao and B. Shu -- Irreducible representations of the special algebras in prime characteristic