**Author**: Maciej Dunajski

**Publisher:** Oxford University Press

**ISBN:**

**Category:** Mathematics

**Page:** 359

**View:** 383

The book provides a self-contained and accessible introduction to elementary twistor theory; a technique for solving differential equations in applied mathematics and theoretical physics. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basicdifferential geometry, complex manifold theory and the exterior differential system.

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

Beginning with the concept of random processes and Brownian motion and building on the theory and research directions in a self-contained manner, this book provides an introduction to stochastic analysis for graduate students, researchers and applied scientists interested in stochastic processes and their applications.

Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration. The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces. The main body of the text is an introduction to the theory of Banach algebras. A particular feature is the detailed account of the holomorphic functional calculus in one and several variables; all necessary background theory in one and several complex variables is fully explained, with many examples and applications considered. Throughout, exercises at sections ends help readers test their understanding, while extensive notes point to more advanced topics and sources. The book was edited for publication by Professor H. G. Dales of Leeds University, following the death of the author in August, 2007.

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.

All papers were peer reviewed. The volume contains reports on work done in areas of field theory, supersymmetry, string theory, higher spins, and related topics using group-theoretical and geometrical methods which involve, one way or another, twistor-like techniques (singletons, harmonics, superembeddings, and twistors themselves).

Twistor theory has become a diverse subject as it has spread from its origins in theoretical physics to applications in pure mathematics. This 1990 collection of review articles covers the considerable progress made in a wide range of applications such as relativity, integrable systems, differential and integral geometry and representation theory. The articles explore the wealth of geometric ideas which provide the unifying themes in twistor theory, from Penrose's quasi-local mass construction in relativity, to the study of conformally invariant differential operators, using techniques of representation theory.

The volume is very beneficial to both starting and experienced researchers working in the field of integrable nonlinear equations, soliton theory, and nonlinear waves. It will be an excellent reference book for graduate students majoring in mathematical physics and engineering sciences. This volume covers a broad range of current interesting topics in nonlinear and modern mathematical physics, and reviews recent developments in integrable systems, soliton theory and nonlinear dynamics. The book is suitable for both starting and experienced researchers working in nonlinear sciences, and it is a good reference for students of mathematical, physical and engineering sciences.