Bücher der Reihe Modern Birkhäuser Classics

A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then It¿s change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman¿Kac functional and the Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.New to the second edition are a discussion of the Cameron¿Martin¿Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.This book willbe a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory. ¿Journal of the American Statistical Association An attractive text¿written in [a] lean and precise style¿eminently readable. Especially pleasant are the care and attention devoted to details¿ A very fine book.¿Mathematical Reviews

Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "e;Queen of Mathematics."e; Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.Andr Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work - from an Old Babylonian tablet, datable to the time of Hammurapi to Legendre's Essai sur la Thorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

This volume contains the first two out of four chapters which are intended to survey a large part of the theory of theta functions. These notes grew out of a series of lectures given at the Tata Institute of Fundamental Research in the period October, 1978, to March, 1979, on which notes were taken and excellently written up by C. Musili and M. Nori. I subsequently lectured at greater length on the contents of Chapter III at Harvard in the fall of 1979 and at a Summer School in Montreal in August, 1980, and again notes were very capably put together by E. Previato and M. Stillman, respectively. Both the Tata Institute and the University of Montreal publish lecture note series in which I had promised to place write-ups of my lectures there. However, as the project grew, it became clear that it was better to tie all these results together, rearranging and consolidating the material, and to make them available from one place. I am very grateful to the Tata Institute and the University of Montreal for permission to do this, and to Birkhauser-Boston for publishing the final result. The first 2 chapters study theta functions strictly from the viewpoint of classical analysis. In particular, in Chapter I, my goal was to explain in the simplest cases why the theta functions attracted attention.

This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kahler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac's quantization of the electrical charge.

The purpose of the present book is to offer an up-to-date account of the theory of viscosity solutions of first order partial differential equations of Hamilton-Jacobi type and its applications to optimal deterministic control and differential games. The theory of viscosity solutions, initiated in the early 80's by the papers of M.G. Crandall and P.L. Lions [CL81, CL83], M.G. Crandall, L.C. Evans and P.L. Lions [CEL84] and P.L. Lions' influential monograph [L82], provides an - tremely convenient PDE framework for dealing with the lack of smoothness of the value functions arising in dynamic optimization problems. The leading theme of this book is a description of the implementation of the viscosity solutions approach to a number of significant model problems in op- real deterministic control and differential games. We have tried to emphasize the advantages offered by this approach in establishing the well-posedness of the c- responding Hamilton-Jacobi equations and to point out its role (when combined with various techniques from optimal control theory and nonsmooth analysis) in the important issue of feedback synthesis.

All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "e;part I'' we find the foundational material, the basic definitions and theorems. In "e;part II"e; we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analy- sis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible conse- quences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com- plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond.

More than twenty years ago I gave a course on Fourier Integral Op­ erators at the Catholic University of Nijmegen (1970-71) from which a set of lecture notes were written up; the Courant Institute of Mathematical Sciences in New York distributed these notes for many years, but they be­ came increasingly difficult to obtain. The current text is essentially a nicely TeXed version of those notes with some minor additions (e.g., figures) and corrections. Apparently an attractive aspect of our approach to Fourier Integral Operators was its introduction to symplectic differential geometry, the basic facts of which are needed for making the step from the local definitions to the global calculus. A first example of the latter is the definition of the wave front set of a distribution in terms of testing with oscillatory functions. This is obviously coordinate-invariant and automatically realizes the wave front set as a subset of the cotangent bundle, the symplectic manifold in which the global calculus takes place.