Series: Cambridge Studies in Advanced Mathematics (No. 119)
Hardback (ISBN-13: 9780521192439)
Non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers are fundamental, is a fast-growing discipline widely used not just within pure mathematics, but also applied in other sciences, including physics, biology and chemistry. This book is the first to provide a comprehensive treatment of non-Archimedean locally convex spaces. The authors provide a clear exposition of the basic theory, together with complete proofs and new results from the latest research. A guide to the many illustrative examples provided, end-of-chapter notes and glossary of terms all make this book easily accessible to beginners at the graduate level, as well as specialists from a variety of disciplines.
* The first book to offer complete coverage of non-Archimedean local convexity
* Readily accessible to graduate students and interested researchers from
various disciplines * Assumes only a basic background in linear algebra,
analysis and topology
Preface; 1. Ultrametrics and valuations; 2. Normed spaces; 3. Locally convex spaces; 4. The Hahn-Banach Theorem; 5. The weak topology; 6. C-compactness; 7. Barrelledness and reflexivity; 8. Montel and nuclear spaces; 9. Spaces with an 'orthogonal' base; 10. Tensor products; 11. Inductive limits; A. Glossary of terms; B. Guide to the examples; Bibliography; Notations; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 120)
Hardback (ISBN-13: 9780521876070)
Rough path analysis provides a fresh perspective on Itofs important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShanefs counterexample) receive eobviousf rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
* A modern introduction made accessible to researchers from related fields
* Provides many exercises and solutions to test the readerfs understanding
* Emphasizes applications to stochastic analysis and interactions with
other areas of mathematics
Preface; Introduction; The story in a nutshell; Part I. Basics: 1. Continuous paths of bounded variation; 2. Riemann-Stieltjes integration; 3. Ordinary differential equations (ODEs); 4. ODEs: smoothness; 5. Variation and Holder spaces; 6. Young integration; Part II. Abstract Theory of Rough Paths: 7. Free nilpotent groups; 8. Variation and Holder spaces on free groups; 9. Geometric rough path spaces; 10. Rough differential equations (RDEs); 11. RDEs: smoothness; 12. RDEs with drift and other topics; Part III. Stochastic Processes Lifted to Rough Paths: 13. Brownian motion; 14. Continuous (semi)martingales; 15. Gaussian processes; 16. Markov processes; Part IV. Applications to Stochastic Analysis: 17. Stochastic differential equations and stochastic flows; 18. Stochastic Taylor expansions; 19. Support theorem and large deviations; 20. Malliavin calculus for RDEs; Part V. Appendix: A. Sample path regularity and related topics; B. Banach calculus; C. Large deviations; D. Gaussian analysis; E. Analysis on local Dirichlet spaces; Frequently used notation; References; Index.
Hardback (ISBN-13: 9780521199704)
Paperback (ISBN-13: 9780521136594)
Do you want a rigorous book that remembers where PDEs come from and what
they look like* This highly visual introduction to linear PDEs and initial/boundary
value problems connects the math to physical reality, all the time providing
a rigorous mathematical foundation for all solution methods. Readers are
gradually introduced to abstraction * the most powerful tool for solving
problems * rather than simply drilled in the practice of imitating solutions
to given examples. The book is therefore ideal for students in mathematics
and physics who require a more theoretical treatment than given in most
introductory texts. Also designed with lecturers in mind, the fully modular
presentation is easily adapted to a course of one-hour lectures, and a
suggested 12-week syllabus is included to aid planning. Downloadable files
for the hundreds of figures, hundreds of challenging exercises, and practice
problems that appear in the book are available online, as are solutions.
* Online resources include full-colour and three-dimensional illustrations,
practice problems and complete solutions for instructors * Includes a suggested
twelve-week syllabus and lists recommended prerequisites for each section
* Contains nearly 400 challenging theoretical exercises
Preface; Notation; What's good about this book*; Suggested twelve-week
syllabus; Part I. Motivating Examples and Major Applications: 1. Heat and
diffusion; 2. Waves and signals; 3. Quantum mechanics; Part II. General
Theory: 4. Linear partial differential equations; 5. Classification of
PDEs and problem types; Part III. Fourier Series on Bounded Domains: 6.
Some functional analysis; 7. Fourier sine series and cosine series; 8.
Real Fourier series and complex Fourier series; 9. Mulitdimensional Fourier
series; 10. Proofs of the Fourier convergence theorems; Part IV. BVP Solutions
Via Eigenfunction Expansions: 11. Boundary value problems on a line segment;
12. Boundary value problems on a square; 13. Boundary value problems on
a cube; 14. Boundary value problems in polar coordinates; 15. Eigenfunction
methods on arbitrary domains; Part V. Miscellaneous Solution Methods: 16.
Separation of variables; 17. Impulse-response methods; 18. Applications
of complex analysis; Part VI. Fourier Transforms on Unbounded Domains:
19. Fourier transforms; 20. Fourier transform solutions to PDEs; Appendices;
References; Index.
With a new Introduction by Mark Ronan
256 pages, 4 line drawings 6 x 9 c 2009
ISBN: 9780226724997
Will publish October 2009
In mathematics, gbuildingsh are geometric structures that represent groups of Lie type over an arbitrary field. This concept is critical to physicists and mathematicians working in discrete mathematics, simple groups, and algebraic group theory, to name just a few areas.
Almost twenty years after its original publication, Mark Ronanfs Lectures on Buildings remains one of the best introductory texts on the subject. A thorough, concise introduction to mathematical buildings, it contains problem sets and an excellent bibliography that will prove invaluable to students new to the field. Lectures on Buildings will find a grateful audience among those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups.
gRonanfs account of the classification of affine buildings [is] both
interesting and stimulating, and his book is highly recommended to those
who already have some knowledge and enthusiasm for the theory of buildings.h*Bulletin
of the London Mathematical Society
ESI Lectures in Mathematics and Physics, vol.6
ISBN 978-3-03719-053-1
June 2009, 307 pages, softcover, 17 x 24 cm.
The general theory of relativity is a theory of manifolds equipped with
Lorentz metrics and fields which describe the matter content. Einsteinfs
equations equate the Einstein tensor (a curvature quantity associated with
the Lorentz metric) with the stress energy tensor (an object constructed
using the matter fields). In addition, there are equations describing the
evolution of the matter. Using symmetry as a guiding principle, one is
naturally led to the Schwarzschild and Friedmann*Lemaitre*Robertson*Walker
solutions, modelling an isolated system and the entire universe respectively.
In a different approach, formulating Einsteinfs equations as an initial
value problem allows a closer study of their solutions. This book first
provides a definition of the concept of initial data and a proof of the
correspondence between initial data and development. It turns out that
some initial data allow non-isometric maximal developments, complicating
the uniqueness issue. The second half of the book is concerned with this
and related problems, such as strong cosmic censorship.
The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those wishing to enter the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.
ISBN : 9782705668518
Un grand nombre des evolutions en theorie des champs, liees au developpement de la physique et des mathematiques actuelles, sont en general caracterisees par la presence des systemes integrables.
La theorie quantique des champs constitue l'une des pierres angulaires de la physique theorique moderne. Ces theories decrivent des systemes de plusieurs particules et possedent en general un grand nombre (souvent infini !) de degres de liberte. Pour cette raison, ils ne peuvent pas etre traites exactement mais plutot en utilisant des methodes perturbatives.
Le concept d'integrabilite s'est avere tres puissant a ce propos. Les developpements dans l'etude des systemes integrables depuis les annees 1970 ont ete a l'origine motives par des problemes physiques concrets, ils ont ensuite conduit a des concepts mathematiques puissants tels que les groupes quantiques dans le contexte de la theorie des champs integrable massive et a une comprehension plus profonde des algebres de Virasoro a la limite de masse nulle, qui sont la plupart du temps des theories conformes.
Dans cet ouvrage, les auteurs explorent les diverses facettes de ces relations profondes entre les theories quantiques des champs et des systemes integrables.
Le livre contient des textes introductifs aux sujets ainsi que des articles plus avances. Il s'adresse a des etudiants en master physique, mathematiques et physique-mathematiques ainsi qu'aux chercheurs interesses par ces questions.
Publication details
Hardcover. 394 pages
plus 40 pages of color and black-white photos.
ISBN-13: 978-1-57146-137-7
To be published: 15 June 2009 (estim.)
Full description
Index Theory is one of the most exciting and consequential accomplishments
of twentieth-century mathematics. The Founders of Index Theory contemplates
the four great mathematicians who developed index theory*Sir Michael Atiyah,
Raoul Bott, Friedrich Hirzebruch, and I.M. Singer*through the eyes of their
students, collaborators and colleagues, their friends and family members,
and themselves.
In addition to their own essays and correspondence*of historical importance*this
volume presents a variety of material of a decidedly personal as well as
compelling mathematical nature, written by some of their most notable students
and long-time collaborators, including such leading current figures in
mathematics and physics as Simon Donaldson, Edward Witten, and S.T. Yau.
In these writings, one perceives the expansive influence of their work
across various fields of mathematics and into theoretical physics.
At a time when the long and illustrious careers of Atiyah, Hirzebruch,
and Singer are being recognized with birthday celebrations at Edinburgh,
in Europe, and in Massachusetts, this second edition of Founders of Index
Theory remembers the late and much beloved Raoul Bott*in the affectionate
words of those three men, as well as family members and long-time friends
and colleagues. What emerges is the portrait of a compelling mathematical
mind informed by a warm and magnetic personality that was both a joy and
inspiration to those who knew him.
This volume includes a generous collection of color and black-and-white
photographs*many rarely seen*of the four principal figures together with
their family, friends, and colleagues. These include numerous images of
Bott dating from his early childhood to his last years at Harvard University.
The Founders of Index Theory, Second Edition is a valuable portrayal of four men who transformed mathematics in a profound manner, and who belong to a class of researchers whose interest and influence transcend the conventional boundaries of mathematical fields.