Edited by: Jean-Marie De Koninck, Universite Laval, Quebec, QC, Canada, Andrew Granville, Universite de Montreal, QC, Canada, and Florian Luca, Universidad Nacional Autonoma de Mexico, Morelia, Mexico

Anatomy of Integers

CRM Proceedings & Lecture Notes, Volume: 46
2008; 297 pp; softcover
ISBN-13: 978-0-8218-4406-9
Expected publication date is August 24, 2008.

The book is mostly devoted to the study of the prime factors of integers, their size and their quantity, to good bounds on the number of integers with different properties (for example, those with only large prime factors) and to the distribution of divisors of integers in a given interval. In particular, various estimates concerning smooth numbers are developed. A large emphasis is put on the study of additive and multiplicative functions as well as various arithmetic functions such as the partition function. More specific topics include the Erd?s-Kac Theorem, cyclotomic polynomials, combinatorial methods, quadratic forms, zeta functions, Dirichlet series and $L$-functions. All these create an intimate understanding of the properties of integers and lead to fascinating and unexpected consequences. The volume includes contributions from leading participants in this active area of research, such as Kevin Ford, Carl Pomerance, Kannan Soundararajan and Gerald Tenenbaum.

Titles in this series are co-published with the Centre de Recherches Mathematiques.


Undergraduates and graduate students and research mathematicians interested in Erd?s-type elementary number theory, smooth numbers, and distribution of prime factors of integers, partitions, etc.

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Edited by: Kristin E. Lauter, Microsoft Research, Redmond, WA,
and Kenneth A. Ribet, University of California at Berkeley, CA

Computational Arithmetic Geometry

Contemporary Mathematics, Volume: 463
2008; 129 pp; softcover
ISBN-13: 978-0-8218-4320-8
Expected publication date is August 30, 2008.

With the recent increase in available computing power, new computations are possible in many areas of arithmetic geometry. To name just a few examples, Cremona's tables of elliptic curves now go up to conductor 120,000 instead of just conductor 1,000, tables of Hilbert class fields are known for discriminant up to at least 5,000, and special values of Hilbert and Siegel modular forms can be calculated to extremely high precision. In many cases, these experimental capabilities have led to new observations and ideas for progress in the field. They have also led to natural algorithmic questions on the feasibility and efficiency of many computations, especially for the purpose of applications in cryptography. The AMS Special Session on Computational Arithmetic Geometry, held on April 29-30, 2006, in San Francisco, CA, gathered together many of the people currently working on the computational and algorithmic aspects of arithmetic geometry. This volume contains research articles related to talks given at the session. The majority of articles are devoted to various aspects of arithmetic geometry, mainly with a computational approach.


Graduate students and research mathematicians interested in arithmetic geometry and computational number theory.

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Edited by: Valery Alexeev, University of Georgia, Athens, GA, Arnaud Beauville, Universite de Nice, Cedex, France, C. Herbert Clemens, Ohio State University, Columbus, OH, and Elham Izadi, University of Georgia, Athens, GA

Curves and Abelian Varieties

Contemporary Mathematics, Volume: 465
2008; 274 pp; softcover
ISBN-13: 978-0-8218-4334-5
Expected publication date is August 17, 2008.

This book is devoted to recent progress in the study of curves and abelian varieties. It discusses both classical aspects of this deep and beautiful subject as well as two important new developments, tropical geometry and the theory of log schemes.

In addition to original research articles, this book contains three surveys devoted to singularities of theta divisors, of compactified Jacobians of singular curves, and of "strange duality" among moduli spaces of vector bundles on algebraic varieties.


Graduate students and research mathematicians interested in algebraic geometry.

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Paul J. Sally, Jr., University of Chicago, IL

Tools of the Trade: Introduction to Advanced Mathematics

2008; 193 pp; hardcover
ISBN-13: 978-0-8218-4634-6
Expected publication date is September 24, 2008.

This book provides a transition from the formula-full aspects of the beginning study of college level mathematics to the rich and creative world of more advanced topics. It is designed to assist the student in mastering the techniques of analysis and proof that are required to do mathematics.

Along with the standard material such as linear algebra, construction of the real numbers via Cauchy sequences, metric spaces and complete metric spaces, there are three projects at the end of each chapter that form an integral part of the text. These projects include a detailed discussion of topics such as group theory, convergence of infinite series, decimal expansions of real numbers, point set topology and topological groups. They are carefully designed to guide the student through the subject matter. Together with numerous exercises included in the book, these projects may be used as part of the regular classroom presentation, as self-study projects for students, or for Inquiry Based Learning activities presented by the students.


Undergraduate and graduate students interested in studying advanced mathematics.

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Sets, functions, and other basic ideas
Linear algebra
The construction of the real and complex numbers
Metric and Euclidean spaces
Complete metric spaces and the p-adic completion of mathbb{Q}

Kenji Ueno, Kyoto University, Japan

Conformal Field Theory with Gauge Symmetry

Fields Institute Monographs, Volume: 24
2008; 168 pp; hardcover
ISBN-13: 978-0-8218-4088-7
Expected publication date is September 5, 2008.

This book presents a systematic approach to conformal field theory with gauge symmetry from the point of view of complex algebraic geometry. After presenting the basic facts of the theory of compact Riemann surfaces and the representation theory of affine Lie algebras in Chapters 1 and 2, conformal blocks for pointed Riemann surfaces with coordinates are constructed in Chapter 3. In Chapter 4 the sheaf of conformal blocks associated to a family of pointed Riemann surfaces with coordinates is constructed, and in Chapter 5 it is shown that this sheaf supports a projective flat connection--one of the most important facts of conformal field theory. Chapter 6 is devoted to the study of the detailed structure of the conformal field theory over $\mathbb{P}^1$.

Recently it was shown that modular functors can be constructed from conformal field theory, giving an interesting relationship between algebraic geometry and topological quantum field theory. This book provides a timely introduction to an intensively studied topic of conformal field theory with gauge symmetry by a leading algebraic geometer, and includes all the necessary techniques and results that are used to construct the modular functor.


Graduate students and research mathematicians interested in algebraic/arithmetic geometry, theoretical physics (high energy) string theory.

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