Edited by: Joseph M. Rosenblatt, University of Illinois at Urbana-Champaign, IL,
and Alexander M. Stokolos and Ahmed I. Zayed, DePaul University, Chicago, IL

Topics in Harmonic Analysis and Ergodic Theory

Contemporary Mathematics, Volume: 444
2007; 228 pp; softcover
ISBN-10: 0-8218-4235-8
ISBN-13: 978-0-8218-4235-5
Expected publication date is November 15, 2007.

There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to problems in number theory.

Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2-4, 2005). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness theorems for multi-dimensional trigonometric series, and Bellman and s-functions.

In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on multi-dimensional trigonometric series.

Readership

Research mathematicians interested in harmonic analysis, ergodic theory, and their interaction.

Table of Contents

• A. I. Zayed -- Topics in ergodic theory and harmonic analysis: An overview
• J. Rosenblatt -- The mathematical work of Roger Jones
• Y. Derriennic and M. Lin -- The central limit theorem for random walks on orbits of probability preserving transformations
• R. F. Gundy -- Probability, ergodic theory, and low-pass filters
• D. J. Rudolph -- Ergodic theory on Borel foliations by mathbb{R}^n and mathbb{Z}^n
• G. V. Welland -- Short review of the work of Professor J. Marshall Ash
• J. M. Ash and G. Wang -- Uniqueness questions for multiple trigonometric series
• C. Fefferman -- Smooth interpolation of functions on mathbb{R}^n
• P. A. Hagelstein -- Problems in interpolation theory related to the almost everywhere convergence of Fourier series
• M. T. Lacey -- Lectures on Nehari's theorem on the polydisk
• L. Slavin and A. Volberg -- The s-function and the exponential integral

William Barker, Bowdoin College, Brunswick, ME, and Roger Howe, Yale University, New Haven, CT

Continuous Symmetry: From Euclid to Klein

2007; 544 pp; hardcover
ISBN-10: 0-8218-3900-4
ISBN-13: 978-0-8218-3900-3
Expected publication date is December 19, 2007.

The fundamental idea of geometry is that of symmetry. With that principle as the starting point, Barker and Howe begin an insightful and rewarding study of Euclidean geometry.

The primary focus of the book is on transformations of the plane. The transformational point of view provides both a path for deeper understanding of traditional synthetic geometry and tools for providing proofs that spring from a consistent point of view. As a result, proofs become more comprehensible, as techniques can be used and reused in similar settings.

The approach to the material is very concrete, with complete explanations of all the important ideas, including foundational background. The discussions of the nine-point circle and wallpaper groups are particular examples of how the strength of the transformational point of view and the care of the authors' exposition combine to give a remarkable presentation of topics in geometry.

This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises.

Readership

Undergraduates interested in geometry.

Table of Contents

• Foundations of geometry in the plane
• Isometries in the plane: Products of reflections
• Isometries in the plane: Classification and structure
• Similarities in the plane
• Conjugacy and geometric equivalence
• Applications to plane geometry
• Symmetric figures in the plane
• Frieze and wallpaper groups
• Area, volume, and scaling
• References
• Index

Alex Iosevich, University of Missouri, Columbia, MO

A View from the Top: Analysis, Combinatorics and Number Theory

Student Mathematical Library, Volume: 39
2007; 136 pp; softcover
ISBN-10: 0-8218-4397-4
ISBN-13: 978-0-8218-4397-0
Expected publication date is December 14, 2007.

This book is based on a capstone course that the author taught to upper division undergraduate students with the goal to explain and visualize the connections between different areas of mathematics and the way different subject matters flow from one another. In teaching his readers a variety of problem solving techniques as well, the author succeeds in enhancing the readers' hands on knowledge of mathematics and provides glimpses into the world of research and discovery. The connections between different techniques and areas of mathematics are emphasized throughout and constitute one of the most important lessons this book attempts to impart. This book is interesting and accessible to anyone with a basic knowledge of high school mathematics and a curiosity about research mathematics.

The author is a professor at the University of Missouri and has maintained a keen interest in teaching at different levels since his undergraduate days at the University of Chicago. He has run numerous summer programs in mathematics for local high school students and undergraduate students at his university. The author gets much of his research inspiration from his teaching activities and looks forward to exploring this wonderful and rewarding symbiosis for years to come.

Readership

Undergraduate students interested in analysis, combinatorics, number theory, and geometry.

Table of Contents

• The Cauchy-Schwarz inequality
• Projections in mathbb{R}^3--The elephant makes an appearance
• Projections in four dimensions
• Projections and cubes
• Incidences and matrices
• Basics of grids over finite fields
• Besicovitch-Kakeya conjecture in two dimensions
• A gentle entry into higher dimensions
• Some basic counting, probability and a few twists
• A more involved taste of probability
• Oscillatory integrals and fun that lies beyond
• Integer points and a crash course on Fourier analysis
• Return of the Fourier transform
• It is time to say goodbye
• Bibliography

Judith D. Sally, Northwestern University, Evanston, IL, and Paul J. Sally, Jr., University of Chicago, IL

Roots to Research: A Vertical Development of Mathematical Problems

2007; 340 pp; hardcover
ISBN-10: 0-8218-4403-2
ISBN-13: 978-0-8218-4403-8
Expected publication date is December 21, 2007.

Certain contemporary mathematical problems are of particular interest to teachers and students because their origin lies in mathematics covered in the elementary school curriculum and their development can be traced through high school, college, and university-level mathematics. This book is intended to provide a source for the mathematics (from beginning to advanced) needed to understand the emergence and evolution of five of these problems: The Four Numbers Problem, Rational Right Triangles, Lattice Point Geometry, Rational Approximation, and Dissection.

Each chapter begins with the elementary geometry and number theory at the source of the problem, and proceeds (with the exception of the first problem) to a discussion of important results in current research. The introduction to each chapter summarizes the contents of its various sections, as well as the background required.

The book is intended for students and teachers of mathematics from high school through graduate school. It should also be of interest to working mathematicians who are curious about mathematical results in fields other than their own. It can be used by teachers at all of the above-mentioned levels for the enhancement of standard curriculum materials or extra-curricular projects.

Readership

High school students, undergraduate and graduate students, and teachers of all levels interested in mathematics.

Table of Contents

• The four numbers problem
• Rational right triangles and the congruent number problem
• Lattice point geometry
• Rational approximation
• Dissection
• Appendix A. Volume
• Appendix B. Convexity
• Index

Emil Artin

Algebra with Galois Theory

Courant Lecture Notes, Volume: 15
2007; 126 pp; softcover
ISBN-10: 0-8218-4129-7
ISBN-13: 978-0-8218-4129-7
Expected publication date is December 7, 2007.

The present text was first published in 1947 by the Courant Institute of Mathematical Sciences of New York University. Published under the title Modern Higher Algebra. Galois Theory, it was based on lectures by Emil Artin and written by Albert A. Blank. This volume became one of the most popular in the series of lecture notes published by Courant. Many instructors used the book as a textbook, and it was popular among students as a supplementary text as well as a primary textbook. Because of its popularity, Courant has republished the volume under the new title Algebra with Galois Theory.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Readership

Undergraduate students interested in algebra.

Table of Contents

• Groups
• Rings and fields
• Polynomials. Factorization into primes. Ideals
• Solution of the general equation of nth degree. Residue classes. Extension fields. Isomorphisms
• Galois theory
• Polynomials with integral coefficients
• The theory of equations

Yuri I. Manin, Northwestern University, Evanston, IL, and Steklov Mathematical Institute, Moscow, Russia

Mathematics as Metaphor: Selected Essays of Yuri I. Manin

Collected Works, Volume: 20
2007; 236 pp; hardcover
ISBN-10: 0-8218-4331-1
ISBN-13: 978-0-8218-4331-4
Expected publication date is January 5, 2008.

The book includes fifteen essays and an interview. The essays are grouped in three parts: Mathematics; Mathematics and Physics; and Language, Consciousness, and Book reviews. Most of the essays are about some aspects of epistemology and the history of sciences, mainly mathematics, physics, and the history of language. English translations of some of the essays, originally published in Russian, appear for the first time in this selection. One of them is the introduction to the book Computable and Uncomputable, where the idea of a quantum computer was first proposed in 1980. Another is an essay on the mythological trickster figure, where the evolutionary role of manipulative behavior is discussed in connection with the problem of the origin of human language. With the foreword by Freeman Dyson, this book will be of interest to anyone interested in the philosophy and history of mathematics, physics, and linguistics.

Readership

Undergraduates, graduate students, research mathematicians, and general public interested in philosophy and history of mathematics, physics, and linguistics.

Table of Contents

Mathematical knowledge: Internal, social, and cultural aspects
Part I. Mathematics as metaphor
Mathematics as metaphor
Truth, rigour, and common sense
Georg Cantor and his heritage
Godel's theorem
Introduction to the book Computable and uncomputable
Mathematics as profession and vocation
Part II. Mathematics and physics
Mathematics and physics
Interrelations between mathematics and physics
Reflections on arithmetical physics
Part III. Language, consciousness, book reviews
The mythological trickster: A study in psychology and culture theory
On early development of speech and consciousness (phylogeny)
The empty city archetype
Triangle of thoughts by A. Connes, A. Lichnerowicz, and M. P. Schutzenberger
"It is still love"-The Siege by Clara Park
"Good proofs are proofs that make us wiser"-Interview by Martin Aigner and Vasco A. Schmidt
List of publications