Series: Spectrum
Hardback (ISBN-13: 9780883855621)
This is a novel, short, and eminently readable history of mathematics.
Many histories provide a chronological history of the entire subject, which
can sometimes make it difficult to follow the development of a particular
branch over time. Dahan-Delmmedico and Pfeiffer succeed splendidly in tracing
each branch from its beginnings forward. They also give an outstanding
account of how the Arabs not only preserved Greek mathematics, but extended
it in the 800 year period from 400*1200. The large number of informative
illustrations support the text and contribute to what is a great read.
* The birth and development of mathematical activity are placed in their
historical, cultural and economic context * Organized to present the development
of the different branches of mathematics * Contains many illustrations
that support the text
Translator's preface; 1. Landscapes; 2. A moment of rationality: Greece; 3. The constitution of classical algebra; 4. Figures, spaces, and geometries; 5. Limits: from the unconceived to the concept; 6. The concept of function and the development of analysis; 7. At the crossroads of algebra, analysis, and geometry: complex numbers; 8. New objects, new laws: the emergence of algebraic structures.
2009; 293 pp; softcover
ISBN-13: 978-0-8218-4883-8
Expected publication date is August 26, 2009.
There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog.
In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the AMS. The posts from 2008 are being published in two volumes.
This book is Part I of the second-year posts, focusing on ergodic theory, combinatorics, and number theory. Chapter 2 consists of lecture notes from Tao's course on topological dynamics and ergodic theory. By means of various correspondence principles, recurrence theorems about dynamical systems are used to prove some deep theorems in combinatorics and other areas of mathematics. The lectures are as self-contained as possible, focusing more on the "big picture" than on technical details.
In addition to these lectures, a variety of other topics are discussed, ranging from recent developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas-Lehmer test for Mersenne primes. Some selected comments and feedback from blog readers have also been incorporated into the articles.
The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.
Graduate students and research mathematicians interested in mathematics in general with focus on ergodic theory, combinatorics, and number theory.
2009; 292 pp; softcover
ISBN-13: 978-0-8218-4885-2
Expected publication date is September 9, 2009.
There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog.
In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the AMS. The posts from 2008 are being published in two volumes.
This book is Part II of the second-year posts, focusing on geometry, topology, and partial differential equations. The major part of the book consists of lecture notes from Tao's course on the Poincare conjecture and its recent spectacular solution by Perelman. The course incorporates a review of many of the basic concepts and results needed from Riemannian geometry and, to a lesser extent, from parabolic PDE. The aim is to cover in detail the high-level features of the argument, along with selected specific components of that argument, while sketching the remaining elements, with ample references to more complete treatments. The lectures are as self-contained as possible, focusing more on the "big picture" than on technical details.
In addition to these lectures, a variety of other topics are discussed, including expository articles on topics such as gauge theory, the Kakeya needle problem, and the Black-Scholes equation. Some selected comments and feedback from blog readers have also been incorporated into the articles.
The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.
Graduate students and research mathematicians interested in mathematics in general with focus on geometry, topology, and partial differential equations.
Expository articles
The Poincare conjecture
Bibliography
Index
University Lecture Series, Volume: 50
2009; 235 pp; softcover
ISBN-13: 978-0-8218-4893-7
Expected publication date is August 23, 2009.
Modern model theory began with Morley's categoricity theorem: A countable first-order theory that has a unique (up to isomorphism) model in one uncountable cardinal (i.e., is categorical in cardinality) if and only if the same holds in all uncountable cardinals. Over the last 35 years Shelah made great strides in extending this result to infinitary logic, where the basic tool of compactness fails. He invented the notion of an Abstract Elementary Class to give a unifying semantic account of theories in first-order, infinitary logic and with some generalized quantifiers. Zilber developed similar techniques of infinitary model theory to study complex exponentiation.
This book provides the first unified and systematic exposition of this work. The many examples stretch from pure model theory to module theory and covers of Abelian varieties. Assuming only a first course in model theory, the book expounds eventual categoricity results (for classes with amalgamation) and categoricity in excellent classes. Such crucial tools as Ehrenfeucht-Mostowski models, Galois types, tameness, omitting-types theorems, multi-dimensional amalgamation, atomic types, good sets, weak diamonds, and excellent classes are developed completely and methodically. The (occasional) reliance on extensions of basic set theory is clearly laid out. The book concludes with a set of open problems.
Graduate students and research mathematicians interested in logic and model theory.
Part 1. Quasiminimal excellence and complex exponentiation
Combinatorial geometries and infinitary logics
Abstract quasiminimality
Covers of the multiplicative group of mathbb{C}
Part 2. Abstract elementary classes
Abstract elementary classes
Two basic results about L_{omega_1,omega}(Q)
Categoricity implies completeness
A model in aleph_2
Part 3. Abstract elementary classes with arbitrarily large models
Galois types, saturation, and stability
Brimful models
Special, limit and saturated models
Locality and tameness
Splitting and minimality
Upward categoricity transfer
Omitting types and downward categoricity
Unions of saturated models
Life without amalgamation
Amalgamation and few models
Part 4. Categoricity in L_{omega_1,omega}
Atomic AEC
Independence in omega-stable classes
Good systems
Excellence goes up
Very few models implies excellence
Very few models implies amalgamation over pairs
Excellence and *-excellence
Quasiminimal sets and categoricity transfer
Demystifying non-excellence
Appendix A. Morley's omitting types theorem
Appendix B. Omitting types in uncountable models
Appendix C. Weak diamonds
Appendix D. Problems
Bibliography
Index