History of Mathematics, Volume: 30
2006; approx. 343 pp; softcover
ISBN-10: 0-8218-4172-6
ISBN-13: 978-0-8218-4172-3
Emil Artin was one of the great mathematicians of the twentieth
century. He had the rare distinction of having solved two of the
famous problems posed by David Hilbert in 1900. He showed that
every positive definite rational function of several variables
was a sum of squares. He also discovered and proved the Artin
reciprocity law, the culmination of over a century and a half of
progress in algebraic number theory.
Artin had a great influence on the development of mathematics in
his time, both by means of his many contributions to research and
by the high level and excellence of his teaching and expository
writing. In this volume we gather together in one place a
selection of his writings wherein the reader can learn some
beautiful mathematics as seen through the eyes of a true master.
The volume's Introduction provides a short biographical sketch of
Emil Artin, followed by an introduction to the books and papers
included in the volume. The reader will first find three of
Artin's short books, titled The Gamma Function, Galois Theory,
and Theory of Algebraic Numbers, respectively. These are followed
by papers on algebra, algebraic number theory, real fields, braid
groups, and complex and functional analysis. The three papers on
real fields have been translated into English for the first time.
The flavor of these works is best captured by the following quote
of Richard Brauer. "There are a number of books and sets of
lecture notes by Emil Artin. Each of them presents a novel
approach. There are always new ideas and new results. It was a
compulsion for him to present each argument in its purest form,
to replace computation by conceptual arguments, to strip the
theory of unnecessary ballast. What was the decisive point for
him was to show the beauty of the subject to the reader."
Copublished with the London Mathematical Society beginning with
Volume 4. Members of the LMS may order directly from the AMS at
the AMS member price. The LMS is registered with the Charity
Commissioners.
Readership
Advanced undergraduates, graduate students, and research
mathematicians interested in number theory and related topics in
the history of mathematics.
Table of Contents
M. Rosen -- Introduction
Books by Emil Artin
E. Artin -- The Gamma Function
E. Artin -- Galois Theory
E. Artin -- Theory of Algebraic Numbers
Papers by Emil Artin
E. Artin and G. Whaples -- Axiomatic characterization of fields
by the product formula for valuations
E. Artin and G. Whaples -- A note on axiomatic characterization
of fields
E. Artin -- A characterization of the field of real algebraic
numbers
E. Artin and O. Schreier -- The algebraic construction of real
fields
E. Artin and O. Schreier -- A characterization of real closed
fields
E. Artin -- The theory of braids
E. Artin -- Theory of braids
E. Artin -- On the theory of complex functions
E. Artin -- A proof of the Krein-Milman theorem
E. Artin -- The influence of J. H. M. Wedderburn on the
development of modern algebra
University Lecture Series, Volume: 39
2006; 151 pp; softcover
ISBN-10: 0-8218-2983-1
ISBN-13: 978-0-8218-2983-7
Moduli problems in algebraic geometry date back to Riemann's
famous count of the 3g-3 parameters needed to determine a curve
of genus g. In this book, Zariski studies the moduli space of
curves of the same equisingularity class. After setting up and
reviewing the basic material, Zariski devotes one chapter to the
topology of the moduli space, including an explicit determination
of the rare cases when the space is compact. Chapter V looks at
specific examples where the dimension of the generic component
can be determined through rather concrete methods. Zariski's last
chapter concerns the application of deformation theory to the
moduli problem, including the determination of the dimension of
the generic component for a particular family of curves.
An appendix by Bernard Teissier reconsiders the moduli problem
from the point of view of deformation theory. He gives new proofs
of some of Zariski's results, as well as a natural construction
of a compactification of the moduli space.
Readership
Graduate students and research mathematicians interested in
algebraic geometry, especially moduli questions, and
singularities.
Table of Contents
2006; 303 pp; hardcover
ISBN-10: 0-8218-4229-3
ISBN-13: 978-0-8218-4229-4
Character theory is a powerful tool for understanding finite
groups. In particular, the theory has been a key ingredient in
the classification of finite simple groups. Characters are also
of interest in their own right, and their properties are closely
related to properties of the structure of the underlying group.
The book begins by developing the module theory of complex group
algebras. After the module-theoretic foundations are laid in the
first chapter, the focus is primarily on characters. This
enhances the accessibility of the material for students, which
was a major consideration in the writing. Also with students in
mind, a large number of problems are included, many of them quite
challenging.
In addition to the development of the basic theory (using a
cleaner notation than previously), a number of more specialized
topics are covered with accessible presentations. These include
projective representations, the basics of the Schur index,
irreducible character degrees and group structure, complex linear
groups, exceptional characters, and a fairly extensive
introduction to blocks and Brauer characters.
This is a corrected reprint of the original 1976 version, later
reprinted by Dover. Since 1976 it has become the standard
reference for character theory, appearing in the bibliography of
almost every research paper in the subject. It is largely self-contained,
requiring of the reader only the most basic facts of linear
algebra, group theory, Galois theory and ring and module theory.
Readership
Graduate students and research mathematicians interested in
finite groups, character theory, and representation theory.
Table of Contents
Algebras, modules, and representations
Group representations and characters
Characters and integrality
Products of characters
Induced characters
Normal subgroups
T.I. sets and exceptional characters
Brauer's theorem
Changing the field
The Schur index
Projective representations
Character degrees
Character correspondence
Linear groups
Changing the characteristic
Some character tables
Bibliographic notes
References
Index
Contemporary Mathematics, Volume: 417
2006; 360 pp; softcover
ISBN-10: 0-8218-3683-8
ISBN-13: 978-0-8218-3683-5
The subject of symmetric functions began with the work of Jacobi,
Schur, Weyl, Young and others on the Schur polynomials. In the
1950's and 60's, far-reaching generalizations of Schur
polynomials were obtained by Hall and Littlewood (independently)
and, in a different direction, by Jack. In the 1980's, Macdonald
unified these developments by introducing a family of polynomials
associated with arbitrary root systems.
The last twenty years have witnessed considerable progress in
this area, revealing new and profound connections with
representation theory, algebraic geometry, combinatorics, special
functions, classical analysis and mathematical physics. All these
fields and more are represented in this volume, which contains
the proceedings of a conference on "Jack, Hall-Littlewood
and Macdonald polynomials" held at ICMS, Edinburgh, during
September 23-26, 2003.
In addition to new results by leading researchers, the book
contains a wealth of historical material, including brief
biographies of Hall, Littlewood, Jack and Macdonald; the original
papers of Littlewood and Jack; notes on Hall's work by Macdonald;
and a recently discovered unpublished manuscript by Jack (annotated
by Macdonald). The book will be invaluable to students and
researchers who wish to learn about this beautiful and exciting
subject.
Readership
Research mathematicians interested in algebraic combinatorics.
Table of Contents
Part 1. Historic Material
B. D. Sleeman -- Henry Jack 1917-1978
A. O. Morris -- Philip Hall
A. O. Morris -- Dudley Ernest Littlewood
A. O. Morris -- Ian Macdonald
I. G. Macdonald -- The algebra of partitions
D. E. Littlewood -- On certain symmetric functions
H. Jack -- A class of symmetric polynomials with a parameter
H. Jack -- A class of polynomials in search of a definition, or
the symmetric group parametrized
I. G. Macdonald -- Commentary on the previous paper
H. Jack, G. de B. Robinson, and W. N. Everitt -- First letter
from Henry Jack to G. de B. Robinson/Second letter reply from G.
de B. Robinson to Henry Jack/Third letter from W. N. Everitt to G.
de B. Robinson
Part 2. Research Articles
H. Coskun and R. A. Gustafson -- Well-poised Macdonald functions
W_{lambda} and Jackson coefficients omega_lambda on BC_n
J. F. van Diejen -- Asymptotics of multivariate orthogonal
polynomials with hyperoctahedral symmetry
P. Etingof and A. Oblomkov -- Quantization, orbifold cohomology,
and Cherednik algebras
B. Ion and S. Sahi -- Triple groups and Cherednik algebras
M. Kasatani, T. Miwa, A. N. Sergeev, and A. P. Veselov --
Coincident root loci and Jack and Macdonald polynomials for
special values of the parameters
T. H. Koornwinder -- Lowering and raising operators for some
special orthogonal polynomials
V. B. Kuznetsov and E. K. Sklyanin -- Factorization of symmetric
polynomials
E. Langmann -- A method to derive explicit formulas for an
elliptic generalization of the Jack polynomials
M. Lassalle -- A short proof of generalized Jacobi-Trudi
expansions for Macdonald polynomials
A. Okounkov and G. Olshanski -- Limits of BC-type orthogonal
polynomials as the number of variables goes to infinity
E. M. Rains -- A difference-integral representation of
Koornwinder polynomials
M. Schlosser -- Explicit computation of the q,t-Littlewood-Richardson
coefficients
V. P. Spiridonov -- A multiparameter summation formula for
Riemann theta functions
Part 3. Vadim Borisovich Kuznetsov 1963-2005
B. D. Sleeman and E. K. Sklyanin -- Vadim Borisovich Kuznetsov
1963-2005
Contemporary Mathematics, Volume: 418
2006; 178 pp; softcover
ISBN-10: 0-8218-4037-1
ISBN-13: 978-0-8218-4037-5
The book consists of contributions related mostly to public-key
cryptography, including the design of new cryptographic
primitives as well as cryptanalysis of previously suggested
schemes. Most papers are original research papers in the area
that can be loosely defined as "non-commutative cryptography";
this means that groups (or other algebraic structures) which are
used as platforms are non-commutative.
Readership
Graduate students and research mathematicians interested in
algebraic methods in cryptography.
Table of Contents
I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux -- Key
agreement, the Algebraic Eraser^{TM}, and lightweight
cryptography
G. Baumslag, T. Camps, B. Fine, G. Rosenberger, and X. Xu --
Designing key transport protocols using combinatorial group
theory
A. Berenstein and L. Chernyak -- Geometric key establishment
P. Dehornoy -- Using shifted conjugacy in braid-based
cryptography
D. Garber, S. Kaplan, M. Teicher, B. Tsaban, and U. Vishne --
Length-based conjugacy search in the braid group
M. I. Gonzalez Vasco, R. Steinwandt, and J. L. Villar -- Towards
provable security for cryptographic constructions arising from
combinatorial group theory
D. Grigoriev and I. Ponomarenko -- Constructions in public-key
cryptography over matrix groups
A. Groch, D. Hofheinz, and R. Steinwandt -- A practical attack on
the root problem in braid groups
D. Hofheinz and D. Unruh -- An attack on a group-based
cryptoraphic scheme
N. G. Leander -- Algebraic problems in symmetric cryptography:
Two recent results on highly nonlinear functions
E. Lee -- Inverting the Burau and Lawrence-Krammer
representations
V. Shpilrain and A. Ushakov -- A new key exchange protocol based
on the decomposition problem
V. Shpilrain and G. Zapata -- Using the subgroup membership
search problem in public key cryptography