This work is based on an undergraduate course taught at the IAS/Park City Mathematics Institute, on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques. The second part of the book discusses travelling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small...

When information is transmitted, errors are likely to occur. This problem has become increasingly important as tremendous amounts of information are transferred electronically every day. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected.

This text is an invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behaviour of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This tool makes possible new analyses and new results.

We have been curious about numbers - and prime numbers - since antiquity. One notable new direction of the 20th century in the study of primes has been the influx of ideas from probability. The goal of this book is to provide insights into the prime numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness.

This volume is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics at Leningrad University. The goal of the course was to present quantum mechanics and its mathematical context to students in mathematics.