AMS/IP Studies in Advanced Mathematics, Volume: 41
2007; 200 pp; softcover
ISBN-10: 0-8218-4298-6
ISBN-13: 978-0-8218-4298-0
Expected publication date is August 18, 2007.
Coding theory draws on a remarkable selection of mathematical topics, both pure and applied. The various contributions in this volume introduce coding theory and its most recent developments and applications, emphasizing both mathematical and engineering perspectives on the subject. This volume covers four important areas in coding theory: algebraic geometry codes, graph-based codes, space-time codes, and quantum codes. Both students and seasoned researchers will benefit from the extensive and self-contained discussions of the development and recent progress in these areas.
Readership
Research mathematicians interested in coding theory.
Table of Contents
Algebraic geometry codes
M.-C. Kang -- Introduction to algebraic geometry codes
W.-C. W. Li -- Upper and lower bounds for A(q)
W.-C. W. Li -- Elkies' modularity conjecture
H. Maharaj -- Explicit towers and codes
W.-C. W. Li -- Improved algebraic geometry bounds
A. Garcia and H. Stichtenoth -- On the Galois closure of towers
Graph-based codes
N. Boston -- Graph-based codes
New aspects of Reed Muller codes
A. R. Calderbank -- Reed Muller codes and symplectic geometry
Quantum codes
A. Ashikhmin and S. Litsyn -- Foundations of quantum error correction
K. Feng -- A new description of quantum error-correcting codes
Clay Mathematics Monographs, Volume: 3
2007; 521 pp; hardcover
ISBN-10: 0-8218-4328-1
ISBN-13: 978-0-8218-4328-4
For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative.
This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate.
The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article.
The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem.
With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology.
The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas.
Readership
Graduate students and research mathematicians interested in geometry and topology.
Table of Contents
Background from Riemannian geometry and Ricci flow
Preliminaries from Riemannian geometry
Manifolds of non-negative curvature
Basics of Ricci flow
The maximum principle
Convergence results for Ricci flow
Perelman's length function and its applications
A comparison geometry approach to the Ricci flow
Complete Ricci flows of bounded curvature
Non-collapsed results
kappa-non-collapsed ancient solutions
Bounded curvature at bounded distance
Geometric limits of generalized Ricci flows
The standard solution
Ricci flow with surgery
Surgery on a delta-neck
Ricci flow with surgery: The definition
Controlled Ricci flows with surgery
Proof of non-collapsing
Completion of the proof of Theorem 15.9
Completion of the proof of the Poincare conjecture
Finite-time extinction
Completion of the proof of Proposition 18.24
3-manifolds covered by canonical neighborhoods
Bibliography
Index
Clay Mathematics Proceedings, Volume: 7
2007; 256 pp; softcover
ISBN-10: 0-8218-4307-9
ISBN-13: 978-0-8218-4307-9
Expected publication date is August 26, 2007.
Articles in this volume are based on talks given at the Gauss-Dirichlet Conference held in Gottingen on June 20-24, 2005. The conference commemorated the 150th anniversary of the death of C.-F. Gauss and the 200th anniversary of the birth of J.-L. Dirichlet.
The volume begins with a definitive summary of the life and work of Dirichlet and continues with thirteen papers by leading experts on research topics of current interest in number theory that were directly influenced by Gauss and Dirichlet. Among the topics are the distribution of primes (long arithmetic progressions of primes and small gaps between primes), class groups of binary quadratic forms, various aspects of the theory of L-functions, the theory of modular forms, and the study of rational and integral solutions to polynomial equations in several variables.
Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Readership
Graduate students and research mathematicians interested in number theory.
Table of Contents
J. Elstrodt -- The life and work of Gustav Lejeune Dirichlet (1805-1859)
T. D. Browning -- An overview of Manin's conjecture for del Pezzo surfaces
J. Brudern and T. D. Wooley -- The density of integral solutions for pairs of diagonal cubic equations
A. Diaconu and D. Goldfeld -- Second moments of GL_2 automorphic L-functions
J. Funke -- CM points and weight 3/2 modular forms
D. A. Goldston, J. Pintz, and C. Y. Yildirim -- The path to recent progress on small gaps between primes
A. Granville and K. Soundararajan -- Negative values of truncations to L(1,chi)
B. Green -- Long arithmetic progressions of primes
P. Michel and A. Venkatesh -- Heegner points and non-vanishing of Rankin/Selberg L-functions
K. Ono -- Singular moduli generating functions for modular curves and surfaces
P. Salberger -- Rational points of bounded height on threefolds
P. Sarnak -- Reciprocal geodesics
K. Soundararajan -- The fourth moment of Dirichlet L-functions
H. M. Stark -- The Gauss class-number problems
AMS Chelsea Publishing
1986; 317 pp; hardcover
ISBN-10: 0-8218-4372-9
ISBN-13: 978-0-8218-4372-7
Expected publication date is September 28, 2007.
Decomposition theory studies decompositions, or partitions, of manifolds into simple pieces, usually cell-like sets. Since its inception in 1929, the subject has become an important tool in geometric topology. The main goal of the book is to help students interested in geometric topology to bridge the gap between entry-level graduate courses and research at the frontier as well as to demonstrate interrelations of decomposition theory with other parts of geometric topology. With numerous exercises and problems, many of them quite challenging, the book continues to be strongly recommended to everyone who is interested in this subject. The book also contains an extensive bibliography and a useful index of key words, so it can also serve as a reference to a specialist.
Readership
Graduate students and research mathematicians interested in geometric topology.
Table of Contents
Introduction
Preliminaries
The shrinkability criterion
Cell-like decompositions of absolute neighborhood retracts
The cell-like approximation theorem
Shrinkable decompositions
Nonshrinkable decompositions
Applications to manifolds
References
Index
2007; approx. 440 pp; hardcover
ISBN-10: 0-8218-4316-8
ISBN-13: 978-0-8218-4316-1
Expected publication date is November 1, 2007.
The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.
Readership
Undergraduates, graduate students, and research mathematicians interested in mathematics.
Table of Contents
Algebra and arithmetics
Arithmetic and combinatorics
Can a number be approximately rational?
Arithmetical properties of binomial coefficients
On collecting like terms, on Euler, Gauss, and MacDonald, and on missed opportunities
Equations
Equations of degree three and four
Equations of degree five
How many roots does a polynomial have?
Chebyshev polynomials
Geometry of equations
Geometry and topology
Envelopes and singularities
Cusps
Around four vertices
Segments of equal areas
On plane curves
Developable surfaces
Paper sheet geometry
Paper Mobius band
More on paper folding
Straight lines
Straight lines on curved surfaces
Twenty-seven lines
Web geometry
The Crofton formula
Polyhedra
Curvature and polyhedra
Non-inscribable polyhedra
Can one make a tetrahedron out of a cube?
Impossible tilings
Rigidity of polyhedra
Flexible polyhedra
Two surprising topological constructions
Alexander's horned sphere
Cone eversion
On ellipses and ellipsoids
Billiards in ellipses and geodesics on ellipsoids
The Poncelet porism and other closure theorems
Gravitational attraction of ellipsoids
Solutions to selected exercises
Bibliography
Index