The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.

Originally published in 1990.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange s reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigate existence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems."

This work concerns the diffeomorphism groups of 3-manifolds, in particular of elliptic 3-manifolds. These are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, now known to be exactly the closed 3-manifolds that have a finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to its diffeomorphism group is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle.

The main results establish the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens spaces L(m, q) with m at least 3. Additional results imply that for a Haken Seifert-fibered 3 manifold V, the space of Seifert fiberings has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background

The contents of this title include: Badrikian, A.: Prolegomenes au calcul des probabilites dans les Banach; Fernique, X.: Regularite des trajectoires des fonctions aleatoires Gaussiennes; Hoffmann-Jorgensen, Jorgen: Probability in Banach space; and, Kuelbs, J.: The law of the iterated logarithm and related strong convergence theorems for Banach space valued random variables.

Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years. This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality. This new 25th Anniversary Edition introduces brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way-as a highbrow comic book!

The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of differential topology. The book contains not only a general theory, but also some explicit examples together with a number of very concrete applications.

This is a very interesting subject in differential topology, since it shows a beautiful interplay between the usual theory of singularities of differentiable maps and the geometric topology of manifolds.

The new edition of this non-mathematical review of catastrophe theory contains updated results and many new or expanded topics including delayed loss of stability, shock waves, and interior scattering. Three new sections offer the history of singularity and its applications from da Vinci to today, a discussion of perestroika in terms of the theory of metamorphosis, and a list of 93 problems touching on most of the subject matter in the book.

The authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.

This is the first attempt of a systematic study of real Enriques surfaces culminating in their classification up to deformation. Simple explicit topological invariants are elaborated for identifying the deformation classes of real Enriques surfaces. Some of theses are new and can be applied to other classes of surfaces or higher-dimensional varieties. Intended for researchers and graduate students in real algebraic geometry it may also interest others who want to become familiar with the field and its techniques. The study relies on topology of involutions, arithmetics of integral quadratic forms, algebraic geometry of surfaces, and the hyperk hler structure of K3-surfaces. A comprehensive summary of the necessary results and techniques from each of these fields is included. Some results are developed further, e.g., a detailed study of lattices with a pair of commuting involutions and a certain class of rational complex surfaces.

The book is devoted to the geometrical construction of the representations of Lusztig's small quantum groups at roots of unity. These representations are realized as some spaces of vanishing cycles of perverse sheaves over configuration spaces. As an application, the bundles of conformal blocks over the moduli spaces of curves are studied. The book is intended for specialists in group representations and algebraic geometry.

Topology optimization is a relatively new and rapidly expanding field of structural mechanics. It deals with some of the most difficult problems of mechanical sciences but it is also of considerable practical interest, because it can achieve much greater savings than mere cross-section or shape optimization.

This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed.

Integration theory holds a prime position, whether in pure mathematics or in various fields of applied mathematics. It plays a central role in analysis; it is the basis of probability theory and provides an indispensable tool in mathe matical physics, in particular in quantum mechanics and statistical mechanics. Therefore, many textbooks devoted to integration theory are already avail able. The present book by Michel Simonnet differs from the previous texts in many respects, and, for that reason, it is to be particularly recommended. When dealing with integration theory, some authors choose, as a starting point, the notion of a measure on a family of subsets of a set; this approach is especially well suited to applications in probability theory. Other authors prefer to start with the notion of Radon measure (a continuous linear func tional on the space of continuous functions with compact support on a locally compact space) because it plays an important role in analysis and prepares for the study of distribution theory. Starting off with the notion of Daniell measure, Mr. Simonnet provides a unified treatment of these two approaches."

Hodge theory is a standard tool in characterizing differ- ential complexes and the topology of manifolds. This book is a study of the Hodge-Kodaira and related decompositions on manifolds with boundary under mainly analytic aspects. It aims at developing a method for solving boundary value problems. Analysing a Dirichlet form on the exterior algebra bundle allows to give a refined version of the classical decomposition results of Morrey. A projection technique leads to existence and regularity theorems for a wide class of boundary value problems for differential forms and vector fields. The book links aspects of the geometry of manifolds with the theory of partial differential equations. It is intended to be comprehensible for graduate students and mathematicians working in either of these fields.

This book brings together experts in the field to explain the ideas involved in the application of the theory of integrable systems to finding harmonic maps and related geometric objects. It had its genesis in a conference with the same title organised by the editors and held at Leeds in May 1992. However, it is not a conference proceedings, but rather a sequence of invited expositions by experts in the field which, we hope, together form a coherent account of the theory. The editors have added cross-references between articles and have written introductory articles in an effort to make the book self-contained. There are articles giving the points of view of both geometry and mathematical physics. Leeds, England A. P. Fordy October 1993 J. e. Wood Authors' addresses J. Bolton, Dept. of Math. Sciences, Univ. of Durham, South Road, Durham, DHI 3LE, UK A. I. Bobenko, FB Math., Tecbnische Univ., Strasse des 17. Juni. 135, 10623 Berlin, Germany M. Bordemann, Falc. fUr Physik, Albert-Ludwigs'Univ., H. -Herder-Str. 3, 79104 Freiburg, Germany F. E. Burstall, Dept. of Mathematics, Univ. of Bath, Claverton Down, Bath, BA 7 7 AY, UK A. P. Fordy, School of Mathematics, Univ. of Leeds, Leeds, LS2 9JT, UK M. Forger, Falc. fUr Physik, Albert-Ludwigs Univ., H. -Herder-Str. 3, 79104 Freiburg, Germany M. A. Guest, Dept. of Mathematics, Univ. of Rochester, Rochester, NY 14627, USA P. Z. Kobalc, Math. Institute, Univ. of Oxford, 24-29 St.

The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr-dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.

An Outline of a General Theory of Models. Translation of Stabilit tructurelle et Morphog'se.

Since quasi-uniform spaces were defined in 1948, a diverse and widely dispersed literatureconcerning them has emerged. In Quasi-Uniform Spaces, the authors present a comprehensivestudy of these structures, together with the theory of quasi-proximities. In additionto new results unavailable elsewhere, the volume unites fundamental materialheretofore scattered throughout the literature.Quasi-Uniform Spaces shows by example that these structures provide a natural approachto the study of point-set topology. It is the only source for many results related to completeness,and a primary source for the study of both transitive and quasi-metric spaces.Included are H. Junnila's analogue of Tamano's theorem, J. Kofner's result showing thatevery GO space is transitive, and R. Fox's example of a non-quasi-metrizable r-space. Inaddition to numerous interesting problems mentioned throughout the text , 22 formalresearch problems are featured. The book nurtures a radically different viewpoint oftopology , leading to new insights into purely topological problems.Since every topological space admits a quasi-uniformity, the study of quasi-uniformspaces can be seen as no less general than the study of topological spaces. For such study,Quasi-Uniform Spaces is a necessary, self-contained reference for both researchers andgraduate students of general topology . Information is made particularly accessible withthe inclusion of an extensive index and bibliography .

The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines. Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences. * Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals * Carefully explains each topic using illustrative examples and diagrams * Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics * Features a wide range of exercises, enabling readers to consolidate their understanding * Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractal Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

This book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.

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.

In the mid-1980's, Abhay Ashtekar discovered a reformulation of general relativity in terms of what are now called the "new variables". This led to a rapid series of developments, and in 1990 Carlo Rovelli and Lee Smolin published a paper entitled "Loop representation of quantum general relativity", in which, among other things, it was shown that there is a space of solutions to the constraint equations of quantum gravity consisting of link invariants. The discovery of relations between quantum gravity and the theory of knots and links came as quite a surprise, since the topics had been regarded as quite remote from each other. This volume is the proceedings of a workshop held in 1993 at the University of California at Riverside. The purpose of the workshop was to bring together researchers in knot theory and quantum gravity and form more bridges between the two subjects. Most of the talks were given by researchers whose work has significance for both subjects. This volume contains expository papers as well as new results, and should serve as a guide for mathematicians and physicists seeking to understand this rapidly developing area of research.

People have been interested in knots at least since the time of Alexander the Great and his encounter with the Gordian knot. There are famous knot illustrations in the Book of Kells and throughout traditional Islamic art. Lord Kelvin believed that atoms were knots in the ether and he encouraged Tait to compile a talbe of knots about 100 years ago. In recent years, the Jones polynomial has stimulated much interest in possible relationships between knot theory and physics. The book is concerned with the fundamental question of the classification of knots, and more generally the classification of arbitrary (compact) topological objects which can occur in our normal space of physical reality. Professor Hemion explains his classification algorithm - using the method of normal surfaces - in a simple and concise way. The reader is thus shown the relevance of such traditional mathematical objects as the Klein bottle or the hyperbolic plane to this basic classification theory. The Classification of Knots and 3-dimensional Spaces will be of interest to mathematicians, physicists, and other scientists who want to apply this basic classification algorithm to their research in knot theory.

How is a subway map different from other maps? What makes a knot knotted? What makes the Moebius strip one-sided? These are questions of topology, the mathematical study of properties preserved by twisting or stretching objects. In the 20th century topology became as broad and fundamental as algebra and geometry, with important implications for science, especially physics. In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.