Steven J. Miller and Ramin Takloo-Bighash

An Invitation to Modern Number Theory

Cloth | April 2006 | ISBN: 0-691-12060-9
519 pp. | 6 x 9 | 20 line illus.

In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.

Steven J. Miller is an Assistant Professor of Mathematics at Brown University. Ramin Takloo-Bighash is an Assistant Professor of Mathematics at Princeton University.

Endorsements:

"The book provides a much-needed introduction to modern number theory that emphasizes analytic number theory. It should serve remarkably well as an advanced undergraduate textbook and its latter parts would be suitable for a beginning graduate course. Some of the material covered, such as the circle method and random matrix theory, is not readily available elsewhere in book form. These topics provide terrific examples of areas in number theory of great current interest that can be penetrated by students. I would seriously consider using this book in my own classes and recommend it with enthusiasm for highly motivated students."--William Duke, University of California, Los Angeles

"Having this selection of material available in essentially self-contained form is fantastic. Reading the book (or taking a class based on it) might easily decide the future endeavors of many a neophyte mathematician. I have yet to discover a clearer exposition of the works of the circle method. The inclusion of exercises and, especially, of problems for further research and theoretical or numerical exploration is extremely valuable. I would dare to compare the book to Hardy and Wright's classic An Introduction to the Theory of Numbers in that Miller and Takloo-Bighash expose readers to the lively work of number theory, to its proofs, ideas, and methods, assuming only a very modest background."--Eduardo Duenez, University of Texas, San Antonio

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Charles R. MacCluer

Honors Calculus

Cloth | May 2006 | ISBN: 0-691-12533-3
200 pp. | 7 x 10 | 38 line illus.

This is the first modern calculus book to be organized axiomatically and to survey the subject's applicability to science and engineering. A challenging exposition of calculus in the European style, it is an excellent text for a first-year university honors course or for a third-year analysis course. The calculus is built carefully from the axioms with all the standard results deduced from these axioms. The concise construction, by design, provides maximal flexibility for the instructor and allows the student to see the overall flow of the development. At the same time, the book reveals the origins of the calculus in celestial mechanics and number theory.

The book introduces many topics often left to the appendixes in standard calculus textbooks and develops their connections with physics, engineering, and statistics. The author uses applications of derivatives and integrals to show how calculus is applied in these disciplines. Solutions to all exercises (even those involving proofs) are available to instructors upon request, making this book unique among texts in the field.


*Focuses on single variable calculus
*Provides a balance of precision and intuition
*Offers both routine and demanding exercises

Professors: A supplementary Solutions Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://pup.princeton.edu/solutions.html

Charles R. MacCluer is Professor of Mathematics and director of the industrial mathematics program at Michigan State University. His first interest was algebraic number theory but later turned to the more practical disciplines of control theory, signal processing, building science, and industrial problems. He is the author of Industrial Mathematics, Boundary Value Problems and Fourier Expansions, and Calculus of Variations.

Viggo Stoltenberg-Hansen (Editor), Jouko Vaananen (Editor)

Logic Colloquium ’03

Lecture Notes in Logic 24

Summary
A compilation of papers presented at the 2003 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium ’03 includes tutorials and research articles from some of the world’s preeminent logicians. One article is a tutorial on finite model theory and query languages that lie between first order and second order logic. The other articles cover current research topics in all areas of mathematical logic, including Proof Theory, Set Theory, Model Theory, and Computability Theory, and Philosophy.

Details
ISBN: 1-56881-293-0
Year: 2006
Format: Hardcover
Pages: 417

ISBN: 1-56881-294-9
Year: 2006
Format: Paperback
Pages: 412

Nigel J. Cutland (Editor), Mauro Di Nasso (Editor), David A. Ross (Editor)

Nonstandard Methods and Applications in Mathematics

Lecture Notes in Logic 25

Summary

A conference on Nonstandard Methods and Applications in Mathematics (NS2002) was held in Pisa, Italy from June 12-16, 2002. Nonstandard analysis is one of the great achievements of modern applied mathematical logic. In addition to the important philosophical achievement of providing a sound mathematical basis for using infinitesimals in analysis, the methodology is now well established as a tool for both research and teaching, and has become a fruitful field of investigation in its own right. This book is a collection of peer-reviewed papers solicited from some of the participants of this conference with the aim of providing something more timely than a textbook, but less ephemeral than a conventional proceedings. It contains both survey papers and research articles with special consideration for one, “Nonstandard analysis at pre-university level: naive magnitude analysis” in which the author discusses his experience teaching calculus through an infinitesimal approach.

Details
ISBN: 1-56881-291-4
Year: 2006
Format: Hardcover
Pages: 262

ISBN: 1-56881-292-2
Year: 2006
Format: Paperback
Pages: 262

Bernstein, Joseph; Hinich, Vladimir; Melnikov, Anna (Eds.)

Studies in Lie Theory
Dedicated to A. Joseph on his Sixtieth Birthday

Series: Progress in Mathematics, Vol. 243
2006, XXII, 494 p. 2 illus., Hardcover
ISBN: 0-8176-4342-7

About this book

Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject.


Two introductory biographical overviews of Joseph's contributions in classical representation theory (the theory of primitive ideals in semisimple Lie algebras) and quantized representation theory (the study of the quantized enveloping algebra) are followed by 16 research articles covering a number of varied and interesting topics in representation theory.

Contributors: J. Alev; A. Beilinson; A. Braverman; I. Cherednik; J. Dixmier; F. Dumas; P. Etingof; D. Farkas; D. Gaitsgory; F. Ivorra; A. Joseph; D. Joseph; M. Kashiwara; D. Kazhdan; A.A. Kirillov; B. Kostant; S. Kumar; G. Letzter; T. Levasseur; G. Lusztig; L. Makar-Limanov; W. McGovern; M. Nazarov; K-H. Neeb; L.G. Rybnikov; P. Schapira; V. Schechtman; A. Sergeev; J.T. Stafford; Ya. Varshavsky; N. Wallach; and I. Waschkies.

Table of contents

* Preface
* Publications of Anthony Joseph
* Students of Anthony Joseph
* From Denise Joseph
* From Jacques Dixmier: A Recollection of Tony Joseph

Part I: Survey and Review
* W. McGovern: The work of Anthony Joseph in classical representation theory
* D. Farkas and G. Letzter: Quantized representation theory following Joseph

Part II: Research Articles
* J. Alev and F. Dumas: Operateurs differentiels invariants et probleme de Noether
* A. Beilinson: Langlands parameters for Heisenberg modules
* A. Braverman and P. Etingof: Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg?Witten prepotential
* I. Cherednik: Irreducibility of perfect representations of double affine Hecke algebras
* D. Gaitsgory and D. Kazhdan: Algebraic groups over a 2-dimensional local field: Some further constructions
* A. Joseph: Modules with a Demazure flag
* M. Kashiwara, P. Schapira, F. Ivorra, and I. Waschkies: Microlocalization of ind-sheaves
* D. Kazhdan and Ya. Varshavsky: Endoscopic decomposition of certain depth zero representations
* A.A. Kirillov and L.G. Rybnikov: Odd family algebras
* B. Kostant and N. Wallach: Gelfand?Zeitlin theory from the perspective of classical mechanics I
* S. Kumar and K-H. Neeb: Extensions of algebraic groups
* T. Levasseur and J.T. Stafford: Differential operators and cohomology groups on the basic affine space
* G. Lusztig: A q-analogue of an identity of N. Wallach
* L. Makar-Limanov: Centralizers in the quantum plane algebra
* M. Nazarov and A. Sergeev: Centralizer construction of the Yangian of the queer Lie superalgebra
* V. Schechtman: Definitio nova algebroidis verticiani