Clay Mathematics Proceedings, Volume: 19
2014; approx. 181 pp; softcover
ISBN-13: 978-0-8218-9865-9
Expected publication date is October 20, 2014.
The conference to celebrate the resolution of the Poincare conjecture, which is one of the Clay Mathematics Institute's seven Millennium Prize Problems, was held at the Institut Henri Poincare in Paris. Several leading mathematicians gave lectures providing an overview of the conjecture--its history, its influence on the development of mathematics, and, finally, its proof.
This volume contains papers based on the lectures at that conference. Taken together, they form an extraordinary record of the work that went into the solution of one of the great problems of mathematics.
Graduate students and research mathematicians interested in geometry and topology of manifolds.
M. Atiyah -- Geometry in 2, 3 and 4 dimensions
J. Morgan -- 100 Years of Topology: Work Stimulated by Poincare's Approach to Classifying Manifolds
C. T. McMullen -- The Evolution of Geometric Structures on 3-Manifolds
S. K. Donaldson -- Invariants of Manifolds and the Classification Problem
D. Gabai, R. Meyerhoff, and P. Milley -- Volumes of Hyperbolic 3-Manifolds
M. Gromov -- Manifolds: Where do we come from? What are we? Where are we going?
G. Tian -- Geometric Analysis on 4-Manifolds
CRM Monograph Series, Volume: 34
2014; 203 pp; hardcover
ISBN-13: 978-1-4704-1693-5
Expected publication date is October 29, 2014.
This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.
Graduate students and research mathematicians interested in symplectic geometry/topology, Hamiltonian dynamics, and quantum mechanics.
Three wonders of symplectic geometry
C0-rigidity of the Poisson bracket
Quasi-morphisms
Subadditive spectral invariants
Symplectic quasi-states and quasi-measures
Applications of partial symplectic quasi-states
A Poisson bracket invariant of quadruples
Symplectic approximation theory
Geometry of covers and quantum noise
Preliminaries from Morse theory
An overview of Floer theory
Constructing subadditive spectral invariants
Bibliography
Nomenclature
Subject index
Name index
Graduate Studies in Mathematics, Volume: 158
2014; 284 pp; hardcover
ISBN-13: 978-0-8218-8319-8
Expected publication date is October 14, 2014.
This book provides an introduction to the interplay between linear algebra and dynamical systems in continuous time and in discrete time. It first reviews the autonomous case for one matrix A via induced dynamical systems in Rd and on Grassmannian manifolds. Then the main nonautonomous approaches are presented for which the time dependency of A(t) is given via skew-product flows using periodicity, or topological (chain recurrence) or ergodic properties (invariant measures). The authors develop generalizations of (real parts of) eigenvalues and eigenspaces as a starting point for a linear algebra for classes of time-varying linear systems, namely periodic, random, and perturbed (or controlled) systems.
The book presents for the first time in one volume a unified approach via Lyapunov exponents to detailed proofs of Floquet theory, of the properties of the Morse spectrum, and of the multiplicative ergodic theorem for products of random matrices. The main tools, chain recurrence and Morse decompositions, as well as classical ergodic theory are introduced in a way that makes the entire material accessible for beginning graduate students.
Graduate students and research mathematicians interested in matrices and random dynamical systems.
Matrices and linear dynamical systems
Autonomous linear differential and difference equations
Linear dynamical systems in Rd
Chain transitivity for dynamical systems
Linear systems in projective space
Linear systems on Grassmannians
Time-varying matrices and linear skew product systems
Lyapunov exponents and linear skew product systems
Periodic linear and differential and difference equations
Morse decompositions of dynamical systems
Topological linear flows
Tools from ergodic theory
Random linear dynamical systems
Bibliography
Index
Graduate Studies in Mathematics, Volume: 159
2014; 229 pp; hardcover
ISBN-13: 978-1-4704-1849-6
Expected publication date is October 23, 2014.
There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught.
On the geometric side, the book introduces coordinates in projective planes and relates coordinate properties to transitivity properties of certain automorphisms and to configuration conditions. It also classifies higher-dimensional geometries and determines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined.
On the algebraic side, basic properties of alternative algebras are derived, including the classification of alternative division rings. As tools for the study of the geometries, an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included.
Graduate students and research mathematicians interested in nonassociative algebra and projective geometry; physicists interested in division algebras and string theory.
Affine and projective planes
Central automorphisms of projective planes
Coordinates for projective planes
Alternative rings
Configuration conditions
Dimension theory
Projective geometries
Automorphisms of G(V)
Quadratic forms and orthogonal groups
Homogeneous maps
Norms and hermitian matrices
Octonion planes
Projective remoteness planes
Other geometries
Bibliography
Index
Contemporary Mathematics, Volume: 623
2014; 280 pp; softcover
ISBN-13: 978-0-8218-9852-9
Expected publication date is October 3, 2014.
This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13-14, 2012, at Tulane University, New Orleans, Louisiana.
Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac-Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.
Graduate students and research mathematicians interested in representation theory.