(Paperback)
ISBN-13: 978-0-19-920652-0
Estimated publication date: March 2008
460 pages, 234x156 mm
Series: Oxford Graduate Texts in Mathematics
Description
Interesting and wide ranging selection of topics
Emphasis on examples and applications will make this text attractive to a wide readership
Well respected author team
Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introduction to rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kahler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction to the topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists; namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.
Readership: Graduates and researchers in mathematics
Contents
1. Lie Groups and Homogeneous Spaces
2. Minimal Models
3. Manifolds
4. Complex and Symplectic Manifolds
5. Geodesics
6. Curvature
7. G-Spaces
8. Blow-ups and Intersection Products
9. A Florilege of Geometric Applications
Appendices
A. De Rham Forms
B. Spectral Sequences
C. Basic Homotopy Recollections
Series: NATO Science for Peace and Security Series
Subseries: NATO Science for Peace and Security Series B: Physics and Biophysics
2008, XVI, 441 p.,
Hardcover ISBN: 978-1-4020-6962-8
Softcover ISBN: 978-1-4020-6963-5
Due: February 2008
About this book
Physical laws are for the most part expressed in terms of differential equations, and natural classes of these are in the form of conservation laws or of problems of the calculus of variations for an action functional. These problems can generally be posed as Hamiltonian systems, whether dynamical systems on finite dimensional phase space as in classical mechanics, or partial differential equations (PDE) which are naturally of infinitely many degrees of freedom. This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems as well as the theory of Hamiltonian systems in infinite dimensional phase space; these are described in depth in this volume. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations. These lecture notes cover many areas of recent mathematical progress in this field, including the new choreographies of many body orbits, the development of rigorous averaging methods which give hope for realistic long time stability results, the development of KAM theory for partial differential equations in one and in higher dimensions, and the new developments in the long outstanding problem of Arnold diffusion. It also includes other contributions to celestial mechanics, to control theory, to partial differential equations of fluid dynamics, and to the theory of adiabatic invariants. In particular the last several years has seen major progress on the problems of KAM theory and Arnold diffusion; accordingly, this volume includes lectures on recent developments of KAM theory in infinite dimensional phase space, and descriptions of Arnold diffusion using variational methods as well as geometrical approaches to the gap problem. The subjects in question involve by necessity some of the most technical aspects of analysis coming from a number of diverse fields. Before the present volume, there has not been one text nor one course of study in which advanced students or experienced researchers from other areas can obtain an overview and background to enter this research area. This volume offers this, in an unparalleled series of extended lectures encompassing this wide spectrum of topics in PDE and dynamical systems.
Table of contents
Series: Lecture Notes in Mathematics , Vol. 470
2nd rev. ed., 2008, Approx. 85 p., Softcover
ISBN: 978-3-540-77605-5
Due: February 20, 2008
About this book
For this printing of R. Bowen's book, J.-R. Chazottes has retyped it in TeX for easier reading, thereby correcting typos and bibliographic details.
From the Preface by D. Ruelle: "Rufus Bowen has left us a masterpiece of mathematical exposition... Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems."
Table of contents
Preface to the 2nd edition by David Ruelle.- 0. Introduction.- 1.Gibbs measures.- A.Gibbs measures.- B.Ruelle's Perron-Frobenius Theorem.- C.Construction of Gibbs measures.- D.Variational principle.- E.Further properties.- References.- 2.General thermodynamic formalism.- A.Entropy.- B.Pressure.- C.Variational principle.- D.Equilibrium states.- References.- 3.Axiom A diffeomorphisms.- A.Definition.- B. Spectral decomposition.- C.Markov partitions.- D.Symbolic dynamics.- References.- 4.Ergodic theory of Axiom A diffeomorphisms.- A.Equilibrium states for basic sets.- B.The case phi=phi^(u).- C.Attractors and Anosov diffeormorphisms.- References.- Index.
Series: Lecture Notes in Mathematics
Subseries: Fondazione C.I.M.E., Firenze , Vol. 1932
2008, Approx. 365 p., Softcover
ISBN: 978-3-540-77644-4
Due: February 27, 2008
About this book
The lectures gathered in this volume present some of the different aspects of Mathematical Control Theory. Adopting the point of view of Geometric Control Theory and of Nonlinear Control Theory, the lectures focus on some aspects of the Optimization and Control of nonlinear, not necessarily smooth, dynamical systems. Specifically, three of the five lectures discuss respectively: logic-based switching control, sliding mode control and the input to the state stability paradigm for the control and stability of nonlinear systems. The remaining two lectures are devoted to Optimal Control: one investigates the connections between Optimal Control Theory, Dynamical Systems and Differential Geometry, while the second presents a very general version, in a non-smooth context, of the Pontryagin Maximum Principle. The arguments of the whole volume are self-contained and are directed to everyone working in Control Theory. They offer a sound presentation of the methods employed in the control and optimization of nonlinear dynamical systems.
Table of contents
Preface by Paolo Nistri and Gianna Stefani.- Andrei A. Agrachev: Geometry of Optimal Control Problems and Hamiltonian Systems.- A. Stephen Morse: Lecture notes on Logically Switched Dynamical Systems.- Eduardo D. Sontag: Input to State Stability: Basic Concepts and Results.- Hector J. Sussmann: Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints.- Vadim I. Utkin: Sliding Mode Control: Mathematical Tools, Design and Applications.
Series: The IMA Volumes in Mathematics and its Applications , Vol. 148
2008, IX, 167 p., Hardcover
ISBN: 978-0-387-78132-7
Due: March 2008
About this book
Algorithms in algebraic geometry go hand in hand with software packages that implement them. Together they have established the modern field of computational algebraic geometry which has come to play a major role in both theoretical advances and applications. Over the past fifteen years, several excellent general purpose packages for computations in algebraic geometry have been developed, such as, CoCoA, Singular and Macaulay 2. While these packages evolve continiously, incorporating new mathematical advances, they both motivate and demand the creation of new mathematics and smarter algorithms.
This volume reflects the workshop "Software for Algebraic Geometry" held in the week from 23 to 27 October 2006, as the second workshop in the thematic year on Applications of Algebraic Geometry at the IMA. The papers in this volume describe the software packages Bertini, PHClab, Gfan, DEMiCs, SYNAPS, TrIm, Gambit, ApaTools, and the application of Risa/Asir to a conjecture on multiple zeta values. They offer the reader a broad view of current trends in computational algebraic geometry through software development and applications.
Table of contents
Software for numerical algebraic geometry: A paradigm and progress towards its implementation.- PhClab: A MATLAB/Octave interface to PHCpack.- Computing Grobner fans and tropical varieties in Gfan.- On a conjecture for the dimension of the space of the multiple zeta values.- DEMiCs: A software package for computing the mixed volume via dynamic enumeration of all mixed cells.- SYNAPS, a library for dedicated applications in symbolic numeric computing.- Tropical implicitization and mixed fiber polytopes.- Towards a black-box solver for finite games: Computing all equilibria with Gambit and PHCpack.- APATools: A software toolbox for approximate polynomial algebra.-
Series: Lecture Notes in Mathematics , Vol. 523
Originally published under: Sergio A. Albeverio, Raphael J. Hoegh-Krohn
2nd corr. and enlarged ed., 2008, Softcover
ISBN: 978-3-540-76954-5
Due: March 19, 2008
About this book
Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low-dimensional topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory.
The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments since then, an entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.
Table of contents
Preface to the second edition.- Preface to the first edition.- 1.Introduction.- 2.The Fresnel Integral of Functions on a Separable Real Hilbert Spa.- 3.The Feynman Path Integral in Potential Scattering.- 4.The Fresnel Integral Relative to a Non-singular Quadratic Form.- 5.Feynman Path Integrals for the Anharmonic Oscillator.- 6.Expectations with Respect to the Ground State of the Harmonic Oscillator.- 7.Expectations with Respect to the Gibbs State of the Harmonic Oscillator.- 8.The Invariant Quasi-free States.- 9.The Feynman Hystory Integral for the Relativistic Quantum Boson Field.- 10.Some Recent Developments.- 10.1.The infinite dimensional oscillatory integral.- 10.2.Feynman path integrals for polynomially growing potentials.- 10.3.The semiclassical expansio.- 10.4.Alternative approaches to Feynman path integrals.- 10.4.1.Analytic continuation.- 10.4.2.White noise calculus.- 10.5.Recent applications.- 10.5.1.The Schroedinger equation with magnetic fields.- 10.5.2.The Schroedinger equation with time dependent potentials.- 10.5.3 .hase space Feynman path integrals.- 10.5.4.The stochastic Schroedinger equation.- 10.5.5.The Chern-Simons functional integral.- References of the first edition.- References of the second edition.- Analytic index.- List of Notations.