These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.

This book surveys the recent theory of wavelet transforms and its applications in various fields both within mathematics (singular integrals, localization of singularities) and beyond it, in computer vision, the physics of fractals, time-frequency analysis.

The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe matical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ ent mathematical approaches, and must have experience with their inter connections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics," "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics" In spi te of i ts difficulty and immensely rich interrela tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many facetted and always new presentations of the material by M. F."

This book is a textbook for graduate or advanced undergraduate students in mathematics and (or) mathematical physics. It is not primarily aimed, therefore, at specialists (or those who wish to become specialists) in integra tion theory, Fourier theory and harmonic analysis, although even for these there might be some points of interest in the book (such as for example the simple remarks in Section 15). At many universities the students do not yet get acquainted with Lebesgue integration in their first and second year (or sometimes only with the first principles of integration on the real line ). The Lebesgue integral, however, is indispensable for obtaining a familiarity with Fourier series and Fourier transforms on a higher level; more so than by us ing only the Riemann integral. Therefore, we have included a discussion of integration theory - brief but with complete proofs - for Lebesgue measure in Euclidean space as well as for abstract measures. We give some emphasis to subjects of which an understanding is necessary for the Fourier theory in the later chapters. In view of the emphasis in modern mathematics curric ula on abstract subjects (algebraic geometry, algebraic topology, algebraic number theory) on the one hand and computer science on the other, it may be useful to have a textbook available (not too elementary and not too spe cialized) on the subjects - classical but still important to-day - which are mentioned in the title of this book."

The Silvri Workshop was divided into a short summer school and a working conference, producing lectures and research papers on recent developments in stochastic analysis on Wiener space. The topics treated in the lectures relate to the Malliavin calculus, the Skorohod integral and nonlinear functionals of white noise. Most of the research papers are applications of these subjects. This volume addresses researchers and graduate students in stochastic processes and theoretical physics.

Piecewise monotone mappings on an interval provide simple examples of discrete dynamical systems whose behaviour can be very complicated. These notes are concerned with the properties of the iterates of such mappings. The material presented can be understood by anyone who has had a basic course in (one-dimensional) real analysis. The account concentrates on the topological (as opposed to the measure theoretical) aspects of the theory of piecewise monotone mappings. As well as offering an elementary introduction to this theory, these notes also contain a more advanced treatment of the problem of classifying such mappings up to topological conjugacy.

Lectures Given at the 1st 1985 Session of the Centro Internazionale Matematico Estivo, (CIME)

Principles of Analysis: Measure, Integration, Functional Analysis, and Applications prepares readers for advanced courses in analysis, probability, harmonic analysis, and applied mathematics at the doctoral level. The book also helps them prepare for qualifying exams in real analysis. It is designed so that the reader or instructor may select topics suitable to their needs. The author presents the text in a clear and straightforward manner for the readers' benefit. At the same time, the text is a thorough and rigorous examination of the essentials of measure, integration and functional analysis. The book includes a wide variety of detailed topics and serves as a valuable reference and as an efficient and streamlined examination of advanced real analysis. The text is divided into four distinct sections: Part I develops the general theory of Lebesgue integration; Part II is organized as a course in functional analysis; Part III discusses various advanced topics, building on material covered in the previous parts; Part IV includes two appendices with proofs of the change of the variable theorem and a joint continuity theorem. Additionally, the theory of metric spaces and of general topological spaces are covered in detail in a preliminary chapter . Features: Contains direct and concise proofs with attention to detail Features a substantial variety of interesting and nontrivial examples Includes nearly 700 exercises ranging from routine to challenging with hints for the more difficult exercises Provides an eclectic set of special topics and applications About the Author: Hugo D. Junghenn is a professor of mathematics at The George Washington University. He has published numerous journal articles and is the author of several books, including Option Valuation: A First Course in Financial Mathematics and A Course in Real Analysis. His research interests include functional analysis, semigroups, and probability.