newpubs-jun-jul99.qxp 5/25/99 1:05 PM Page 716 New Publications Offered by the AMS New Series from the AMS! The AMS is pleased to announce the Student Mathematical Library, a new series of undergraduate studies in mathematics. By emphasizing original topics and approaches, the series aims to broaden students’ mathematical experiences. We hope the books will spark undergraduates’ appreciation for research mathematics by introducing them to interesting, accessible topics of modern mathematics. The books published in the series are suitable for honors courses, upper-division seminars, reading courses or self-study. Advance Notice Recommended Text Recommended Text STUDENT MATHEMATICAL LIBRARY IAS PARK CITY MATHEMATICAL SUBSERIES Lectures on VOLUME 2 An Introduction to the Contemporary Mathematical Theory of Waves Lectures on Contemporary Probability Roger A. Knobel, University of Texas-Pan Probability Gregory F. Lawler, Duke American, Edinburg Gregory F. Lawler Lester N. Coyle University, Durham, NC, and This book is based on an undergraduate course taught at the Lester N. Coyle, Loyola THEMAT A IC IAS/Park City Mathematics Institute, on linear and nonlinear M A L College, Baltimore, MD N ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ A S O C I C R I E E ΑΓΕΩΜΕ T M Y A F OU 88 waves. The first part of the text overviews the concept of a NDED 18 American Mathematical Society wave, describes one-dimensional waves using functions of two Institute for Advanced Study This volume is based on classes in variables, provides an introduction to partial differential equa- probability for advanced undergradu- tions, and discusses computer-aided visualization techniques. ates held at the IAS/Park City Mathematics Institute. It is derived from both lectures (Chapters 1–10) and computer The second part of the book discusses traveling waves, leading simulations (Chapters 11–13) that were held during the to a description of solitary waves and soliton solutions of the program. The material is coordinated so that some of the Klein-Gordon and Korteweg-deVries equations. The wave equa- major computer simulations relate to topics covered in the tion is derived to model the small vibrations of a taut string, first ten chapters. The goal is to present topics that are acces- and solutions are constructed via d’Alembert’s formula and sible to advanced undergraduates, yet are areas of current Fourier series. research in probability. The combination of the lucid yet The last part of the book discusses waves arising from conser- informal style of the lectures and the hands-on nature of the vation laws. After deriving and discussing the scalar simulations allows readers to become familiar with some inter- conservation law, its solution is described using the method of esting and active areas of probability. characteristics, leading to the formation of shock and rarefac- The first four chapters discuss random walks and the contin- tion waves. Applications of these concepts are then given for uous limit of random walks: Brownian motion. Chapters 5 and models of traffic flow. 6 consider the fascinating mathematics of card shuffles, The intent of this book is to create a text suitable for indepen- including the notions of random walks on a symmetric group dent study by undergraduate students in mathematics, and the general idea of random permutations. engineering, and science. The content of the book is meant to Chapters 7 and 8 discuss Markov chains, beginning with a be self-contained, requiring no special reference material. standard introduction to the theory. Chapter 8 addresses the Access to computer software such as Mathematica®, MATLAB®, recent important application of Markov chains to simulations or Maple® is recommended, but not necessary. Scripts for of random systems on large finite sets: Markov Chain Monte MATLAB applications will be available via a Web site. Exercises Carlo. are given within the text to allow further practice with selected topics. Random walks and electrical networks are covered in Chapter 9. Uniform spanning trees, as connected to probability and ® Wolfram Research, Inc., Champaign IL. random walks, are treated in Chapter 10. ® The Math Works, Inc., Natick, MA. ® Waterloo Maple, Inc., Ontario, Canada. The final three chapters of the book present simulations. Student Mathematical Library Chapter 11 discusses simulations for random walks. Chapter 12 covers simulation topics such as sampling from continuous October 1999 (estimated), approximately 200 pages, Softcover, distributions, random permutations, and estimating the ISBN 0-8218-2039-7, All AMS members $18, List $23, Order number of matrices with certain conditions using Markov code STML-KNOBELN Chain Monte Carlo. Chapter 13 presents simulations of stochastic differential equations for applications in finance. 716 NOTICES OF THE AMS VOLUME 46, NUMBER 6 newpubs-jun-jul99.qxp 5/25/99 1:05 PM Page 717 New Publications Offered by the AMS (The simulations do not require one particular piece of soft- Advance Notice ware. They can be done in symbolic computation packages or Independent Study via programming languages such as C.) The volume concludes with a number of problems ranging Prime Numbers and Their from routine to very difficult. Of particular note are problems that are typical of simulation problems given to students by Distribution the authors when teaching undergraduate probability. Gérald Tenenbaum, Université Henri Poincaré, Contents: Simple random walk and Stirling’s formula; Simple Nancy I, France, and Michel Mendès France, ramdon walk in many dimensions; Self-avoiding walk; Université Bordeaux I, France Brownian motion; Shuffling and random permutations; Seven shuffles are enough (sort of); Markov chains on finite sets; From a review for the French edition … Markov chain Monte Carlo; Random walks and electrical This is a short introductory book on analytic number theory. networks; Uniform spanning trees; Random walk simulations; The prerequisites are quite modest, but it still contains an Other simulations; Simulations in finance; Problems; Bibliog- impressive amount of information. A multitude of results is raphy. included, some of which were proved just recently ... this book is Student Mathematical Library very well written. It is fun to read and at the same time presents most of the fundamental concepts and ideas in analytic number August 1999, approximately 120 pages, Softcover, theory. ISBN 0-8218-2029-X, 1991 Mathematics Subject Classification: —Mathematical Reviews 60–02; 60J10, 60J15, 65C05, All AMS members $14, List $17, Order code STML-LAWLERN The reviewer recommends it to all interested readers. —Zentralblatt für Mathematik Supplementary Reading We have been curious about numbers—and prime numbers— since antiquity. One notable new direction this century in the STUDENT MATHEMATICAL LIBRARY Miles of Tiles VOLUME 1 study of primes has been the influx of ideas from probability. Charles Radin, University of The goal of this book is to provide insights into the prime Texas, Austin numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness. Miles In this book, we try to display the There are two ways in which the book is exceptional. First, of Tiles value (and joy!) of starting from a some familiar topics are covered with refreshing insight Charles Radin mathematically amorphous problem and/or from new points of view. Second, interesting recent and combining ideas from diverse developments and ideas are presented that shed new light on sources to produce new and significant the prime numbers and their distribution among the rest of mathematics—mathematics unfore- the integers. seen from the motivating problem... —from the Preface The book begins with a chapter covering some classic topics, such as quadratic residues and the Sieve of Eratosthenes. Also The common thread throughout this book is aperiodic tilings; discussed are other sieves, primes in cryptography, twin the best-known example is the “kite and dart” tiling. This tiling primes, and more. has been widely discussed, particularly since 1984 when it was adopted to model quasicrystals. The presentation uses many Two separate chapters address the Riemann zeta function and different areas of mathematics and physics to analyze the new its connections to number theory. In the first chapter, the features of such tilings. Although many people are aware of familiar link between (s) and the distribution of primes is the existence of aperiodic tilings, and maybe even their origin covered with remarkable efficiency and intuition. The second in a question in logic, not everyone is familiar with their chapter presents a walk through an elementary proof of the subtleties and the underlying rich mathematical theory. For the Prime Number Theorem. To help the novice understand the interested reader, this book fills that gap. “‘why” of the proof, connections are made along the way with more familiar results such as Stirling’s formula. Understanding this new type of tiling requires an unusual variety of specialties, including ergodic theory, functional A most distinctive chapter covers the stochastic properties of analysis, group theory and ring theory from mathematics, and prime numbers. The authors present a wonderfully clever statistical mechanics and wave diffraction from physics. This interpretation of primes in arithmetic progressions as a interdisciplinary approach also leads to new mathematics phenomenon in probability. They also describe Cramér’s seemingly unrelated to the tilings. Included are many worked