Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

This is a monograph on geometrical and topological features which arise in quantum field theory. It is well known that when a chiral fermion interacts with a gauge field we have chiral anomaly which corresponds to the fact that divergence of the axial vector current does not vanish. It is observed that this is related to certain topological features associated with the fermion and leads to the realization of the topological origin of fermion number as well as the Berry phase. The role of gauge fields in the quantization procedure has its implications in these topological features of a fermion and helps us to consider a massive fermion as a soliton (skyrrnion). In this formalism chiral anomaly is found to be responsible for mass generation. This has its relevance in electroweak theory where it is observed that weak interaction gauge bosons attain mass topologically. The geometrical feature of a skyrmion also helps us to realize the internal symmetry of hadrons from reflection group. Finally it has been shown that noncommutative geometry where the space time manifold is taken to be X = M x Zz has its relevance in the description of a massive 4 fermion as a skyrmion when the discrete space is considered as the internal space and the symmetry breaking leads to chiral anomaly. In chap. l preliminary mathematical formulations related to the spinor structure have been discussed. In chap.

The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.

The so-called boundary element methods BEM, i.e. finite element approxima tions of boundary integral equations have been improved recently even more vividly then ever before and found some remarkable support by the German Research Foundation DFG in the just finished Priority Research Program "boundary element methods" . When this program began, we could start from several already existing particular activities which then during the six years initiated many new re sults and decisive new developments in theory and algorithms. The program was started due to encouragement by E. Stein, when most of the later par ticipants met in Stuttgart at a Boundary Element Conference 1987. Then W. Hackbusch, G. Kuhn, S. Wagner and W. Wendland were entrusted with writing the proposal which was 1988 presented at the German Research Foun dation and started in 1989 with 14 projects at 11 different universities. After German unification, the program was heavily extended by six more projects, four of which located in Eastern Germany. When we started, we were longing for the following goals: 1. Mathematicians and engineers should do joint research. 2. Methods and computational algorithms should be streamlined with re spect to the new computer architectures of vector and parallel computers. 3. The asymptotic error analysis of boundary element methods should be further developed. 4. Non-linear material laws should be taken care of by boundary element methods for crack-mechanics. 5. The coupling of finite boundary elements should be improved."

This book is about the light like (degenerate) geometry of submanifolds needed to fill a gap in the general theory of submanifolds. The growing importance of light like hypersurfaces in mathematical physics, in particular their extensive use in relativity, and very limited information available on the general theory of lightlike submanifolds, motivated the present authors, in 1990, to do collaborative research on the subject matter of this book. Based on a series of author's papers (Bejancu [3], Bejancu-Duggal [1,3], Dug gal [13], Duggal-Bejancu [1,2,3]) and several other researchers, this volume was conceived and developed during the Fall '91 and Fall '94 visits of Bejancu to the University of Windsor, Canada. The primary difference between the lightlike submanifold and that of its non degenerate counterpart arises due to the fact that in the first case, the normal vector bundle intersects with the tangent bundle of the submanifold. Thus, one fails to use, in the usual way, the theory of non-degenerate submanifolds (cf. Chen [1]) to define the induced geometric objects (such as linear connection, second fundamental form, Gauss and Weingarten equations) on the light like submanifold. Some work is known on null hypersurfaces and degenerate submanifolds (see an up-to-date list of references on pages 138 and 140 respectively). Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of an up-to-date information on null curves, lightlike hypersur faces and submanifolds, consistent with the theory of non-degenerate submanifolds.

In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.

Few people outside of mathematics are aware of the varieties of mathemat ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function."

A major flaw in semi-Riemannian geometry is a shortage of suitable types of maps between semi-Riemannian manifolds that will compare their geometric properties. Here, a class of such maps called semi-Riemannian maps is introduced. The main purpose of this book is to present results in semi-Riemannian geometry obtained by the existence of such a map between semi-Riemannian manifolds, as well as to encourage the reader to explore these maps. The first three chapters are devoted to the development of fundamental concepts and formulas in semi-Riemannian geometry which are used throughout the work. In Chapters 4 and 5 semi-Riemannian maps and such maps with respect to a semi-Riemannian foliation are studied. Chapter 6 studies the maps from a semi-Riemannian manifold to 1-dimensional semi- Euclidean space. In Chapter 7 some splitting theorems are obtained by using the existence of a semi-Riemannian map. Audience: This volume will be of interest to mathematicians and physicists whose work involves differential geometry, global analysis, or relativity and gravitation.

Application of the concepts and methods of topology and geometry have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity and particle physics. This book can be considered an advanced textbook on modern applications and recent developments in these fields of physical research. Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, sypersymmetric solitons and noncommutative geometry. It will be of benefit to postgraduate students, educating newcomers to the field and lecturers looking for advanced material.

This monograph is mostly devoted to the problem of the geome- trizing of Lagrangians which depend on higher order accelerations. It naturally prolongs the theme of the monograph "The Geometry of La- grange spaces: Theory and Applications", written together with M. Anastasiei and published by Kluwer Academic Publishers in 1994. The existence of Lagrangians of order k > 1 has been contemplated by mechanicists and physicists for a long time. Einstein had grasped their presence in connection with the Brownian motion. They are also present in relativistic theories based on metrics which depend on speeds and accelerations of particles or in the Hamiltonian formulation of non- linear systems given by Korteweg-de Vries equations. There resulted from here the methods to be adopted in their theoretical treatment. One is based on the variational problem involving the integral action of the Lagrangian. A second one is derived from the axioms of Analytical Mechanics involving the Poincare-Cartan forms. The geometrical methods based on the study of the geometries of higher order could invigorate the whole theory. This is the way adopted by us in defining and studying the Lagrange spaces of higher order. The problems raised by the geometrization of Lagrangians of order k > 1 investigated by many scholars: Ch. Ehresmann, P. Libermann, J. Pommaret; J.T. Synge, M. Crampin, P. Saunders; G.S. Asanov, P.Aringazin; I. Kolar, D. Krupka; M. de Leon, W. Sarlet, P. Cantrjin, H. Rund, W.M. Tulczyjew, A. Kawaguchi, K. Yano, K. Kondo, D.

The central theme of this book is the restoration of Poincare duality on stratified singular spaces by using Verdier-self-dual sheaves such as the prototypical intersection chain sheaf on a complex variety. Highlights include complete and detailed proofs of decomposition theorems for self-dual sheaves, explanation of methods for computing twisted characteristic classes and an introduction to the author's theory of non-Witt spaces and Lagrangian structures."

In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives a comprehensive and systematic account on the Carath odory and Kobayashi distances, hyperbolic complex spaces and holomorphic mappings with geometric methods. A very complete list of references should be useful for prospective researchers in this area.

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details normally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spinor calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spinor techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 - - space-time models with non-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet another book on spinor and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spinor c- culus is the unifying mathematical language.

This volume is devoted to the "hyperbolic theory" of dynamical systems (DS), that is, the theory of smooth DS's with hyperbolic behaviour of the tra jectories (generally speaking, not the individual trajectories, but trajectories filling out more or less "significant" subsets in the phase space. Hyperbolicity the property that under a small displacement of any of a trajectory consists in point of it to one side of the trajectory, the change with time of the relative positions of the original and displaced points resulting from the action of the DS is reminiscent of the mot ion next to a saddle. If there are "sufficiently many" such trajectories and the phase space is compact, then although they "tend to diverge from one another" as it were, they "have nowhere to go" and their behaviour acquires a complicated intricate character. (In the physical literature one often talks about "chaos" in such situations. ) This type of be haviour would appear to be the opposite of the more customary and simple type of behaviour characterized by its own kind of stability and regularity of the motions (these words are for the moment not being used as a strict ter 1 minology but rather as descriptive informal terms). The ergodic properties of DS's with hyperbolic behaviour of trajectories (Bunimovich et al. 1985) have already been considered in Volume 2 of this series. In this volume we therefore consider mainly the properties of a topological character (see below 2 for further details)."

The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology.

The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

In inverse problems, the aim is to obtain, via a mathematical model, information on quantities that are not directly observable but rather depend on other observable quantities. Inverse problems are encountered in such diverse areas of application as medical imaging, remote sensing, material testing, geosciences and financing. It has become evident that new ideas coming from differential geometry and modern analysis are needed to tackle even some of the most classical inverse problems. This book contains a collection of presentations, written by leading specialists, aiming to give the reader up-to-date tools for understanding the current developments in the field.

This book is a comprehensive reference on differential geometry. It shows that Maxwell, Dirac and Einstein fields, which were originally considered objects of a very different mathematical nature, have representatives as objects of the same mathematical nature. The book also analyzes some foundational issues of relativistic field theories. All calculation procedures are illustrated by many exercises that are solved in detail.

The theory of foliations of manifolds was created in the forties of the last century by Ch. Ehresmann and G. Reeb [ER44]. Since then, the subject has enjoyed a rapid development and thousands of papers investigating foliations have appeared. A list of papers and preprints on foliations up to 1995 can be found in Tondeur [Ton97]. Due to the great interest of topologists and geometers in this rapidly ev- ving theory, many books on foliations have also been published one after the other. We mention, for example, the books written by: I. Tamura [Tam76], G. Hector and U. Hirsch [HH83], B. Reinhart [Rei83], C. Camacho and A.L. Neto [CN85], H. Kitahara [Kit86], P. Molino [Mol88], Ph. Tondeur [Ton88], [Ton97], V. Rovenskii [Rov98], A. Candel and L. Conlon [CC03]. Also, the survey written by H.B. Lawson, Jr. [Law74] had a great impact on the de- lopment of the theory of foliations. So it is natural to ask: why write yet another book on foliations? The answerisverysimple.Ourareasofinterestandinvestigationaredi?erent.The main theme of this book is to investigate the interrelations between foliations of a manifold on one hand, and the many geometric structures that the ma- foldmayadmitontheotherhand. Amongthesestructureswemention:a?ne, Riemannian, semi-Riemannian, Finsler, symplectic, and contact structures.

GU Chaohao The soliton theory is an important branch of nonlinear science. On one hand, it describes various kinds of stable motions appearing in - ture, such as solitary water wave, solitary signals in optical ?bre etc., and has many applications in science and technology (like optical signal communication). On the other hand, it gives many e?ective methods ofgetting explicit solutions of nonlinear partial di?erential equations. Therefore, it has attracted much attention from physicists as well as mathematicians. Nonlinearpartialdi?erentialequationsappearinmanyscienti?cpr- lems. Getting explicit solutions is usually a di?cult task. Only in c- tain special cases can the solutions be written down explicitly. However, for many soliton equations, people have found quite a few methods to get explicit solutions. The most famous ones are the inverse scattering method, B] acklund transformation etc.. The inverse scattering method is based on the spectral theory of ordinary di?erential equations. The Cauchyproblemofmanysolitonequationscanbetransformedtosolving a system of linear integral equations. Explicit solutions can be derived when the kernel of the integral equation is degenerate. The B] ac ] klund transformation gives a new solution from a known solution by solving a system of completely integrable partial di?erential equations. Some complicated "nonlinear superposition formula" arise to substitute the superposition principlein linear science."

This is an introduction to classical and quantum mechanics on two-point homogenous Riemannian spaces, empahsizing spaces with constant curvature. Chapters 1-4 provide basic notations for studying two-body dynamics. Chapter 5 deals with the problem of finding explicitly invariant expressions for the two-body quantum Hamiltonian. Chapter 6 addresses one-body problems in a central potential. Chapter 7 investigates the classical counterpart of the quantum system introduced in Chapter 5. Chapter 8 discusses applications in the quantum realm.

Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. Thus the appearance of a book specifically devoted to projective duality is a long-awaited and welcome event.

Projective Duality and Homogeneous Spaces covers a vast and diverse range of topics in the field of dual varieties, ranging from differential geometry to Mori theory and from topology to the theory of algebras. It gives a very readable and thorough account and the presentation of the material is clear and convincing. For the most part of the book the only prerequisites are basic algebra and algebraic geometry.

This book will be of great interest to graduate and postgraduate students as well as professional mathematicians working in algebra, geometry and analysis.

This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

Since 1992 Finsler geometry, Lagrange geometry and their applications to physics and biology, have been intensive1y studied in the context of a 5-year program called "Memorandum ofUnderstanding", between the University of Alberta and "AL.1. CUZA" University in lasi, Romania. The conference, whose proceedings appear in this collection, belongs to that program and aims to provide a forum for an exchange of ideas and information on recent advances in this field. Besides the Canadian and Romanian researchers involved, the conference benefited from the participation of many specialists from Greece, Hungary and Japan. This proceedings is the second publication of our study group. The first was Lagrange Geometry. Finsler spaces and Noise Applied in Biology and Physics (1]. Lagrange geometry, which is concerned with regular Lagrangians not necessarily homogeneous with respect to the rate (i.e. velocities or production) variables, naturalIy extends Finsler geometry to alIow the study of, for example, metrical structures (i.e. energies) which are not homogeneous in these rates. Most Lagrangians arising in physics falI into this class, for example. Lagrange geometry and its applications in general relativity, unified field theories and re1ativistic optics has been developed mainly by R. Miron and his students and collaborators in Romania, while P. Antonelli and his associates have developed models in ecology, development and evolution and have rigorously laid the foundations ofFinsler diffusion theory [1] .

Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities a la Painleve, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables.
The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved."