This volume offers an introduction, in the form of four extensive lectures, to some recent developments in several active topics at the interface between geometry, topology and quantum field theory. The first lecture is by Christine Lescop on knot invariants and configuration spaces, in which a universal finite-type invariant for knots is constructed as a series of integrals over configuration spaces. This is followed by the contribution of Raimar Wulkenhaar on Euclidean quantum field theory from a statistical point of view. The author also discusses possible renormalization techniques on noncommutative spaces. The third lecture is by Anamaria Font and Stefan Theisen on string compactification with unbroken supersymmetry. The authors show that this requirement leads to internal spaces of special holonomy and describe Calabi-Yau manifolds in detail. The last lecture, by Thierry Fack, is devoted to a K-theory proof of the Atiyah-Singer index theorem and discusses some applications of K-theory to noncommutative geometry. These lectures notes, which are aimed in particular at graduate students in physics and mathematics, start with introductory material before presenting more advanced results. Each chapter is self-contained and can be read independently.

The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro't from each other. Although the most striking results have appeared on the mathema- calside, theoreticalphysicshasclearlyalsobene?tted, sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theor

Gaussian scale-space is one of the best understood multi-resolution techniques available to the computer vision and image analysis community. It is the purpose of this book to guide the reader through some of its main aspects. During an intensive weekend in May 1996 a workshop on Gaussian scale-space theory was held in Copenhagen, which was attended by many of the leading experts in the field. The bulk of this book originates from this workshop. Presently there exist only two books on the subject. In contrast to Lindeberg's monograph (Lindeberg, 1994e) this book collects contributions from several scale space researchers, whereas it complements the book edited by ter Haar Romeny (Haar Romeny, 1994) on non-linear techniques by focusing on linear diffusion. This book is divided into four parts. The reader not so familiar with scale-space will find it instructive to first consider some potential applications described in Part 1. Parts II and III both address fundamental aspects of scale-space. Whereas scale is treated as an essentially arbitrary constant in the former, the latter em phasizes the deep structure, i.e. the structure that is revealed by varying scale. Finally, Part IV is devoted to non-linear extensions, notably non-linear diffusion techniques and morphological scale-spaces, and their relation to the linear case. The Danish National Science Research Council is gratefully acknowledged for providing financial support for the workshop under grant no. 9502164."

As K. Nomizu has justly noted [K. Nomizu, 56], Differential Geometry ever will be initiating newer and newer aspects of the theory of Lie groups. This monograph is devoted to just some such aspects of Lie groups and Lie algebras. New differential geometric problems came into being in connection with so called subsymmetric spaces, subsymmetries, and mirrors introduced in our works dating back to 1957 [L.V. Sabinin, 58a,59a,59b]. In addition, the exploration of mirrors and systems of mirrors is of interest in the case of symmetric spaces. Geometrically, the most rich in content there appeared to be the homogeneous Riemannian spaces with systems of mirrors generated by commuting subsymmetries, in particular, so called tri-symmetric spaces introduced in [L.V. Sabinin, 61b]. As to the concrete geometric problem which needs be solved and which is solved in this monograph, we indicate, for example, the problem of the classification of all tri-symmetric spaces with simple compact groups of motions. Passing from groups and subgroups connected with mirrors and subsymmetries to the corresponding Lie algebras and subalgebras leads to an important new concept of the involutive sum of Lie algebras [L.V. Sabinin, 65]. This concept is directly concerned with unitary symmetry of elementary par- cles (see [L.V. Sabinin, 95,85] and Appendix 1). The first examples of involutive (even iso-involutive) sums appeared in the - ploration of homogeneous Riemannian spaces with and axial symmetry. The consideration of spaces with mirrors [L.V. Sabinin, 59b] again led to iso-involutive sums.

The contemporary theoretical physics consists, by and large, of two independent parts. The rst is the quantum theory describing the micro-world of elementary p- ticles, the second is the theory of gravity that concerns properties of macroscopic systems such as stars, galaxies, and the universe. The relativistic theory of gr- itation which is known as general relativity was created, at the beginning of the last century, by more or less a single man from pure idea combinations and bold guessing. The task was to "marry" the theory of gravity with the theory of special relativity. The rst attempts were aimed at considering the gravitational potential as a eld in Minkowski space-time. All those attempts failed; it took 10 years until Einstein nally solved the problem. The dif culty was that the old theory of gravity as well as the young theory of special relativity had to be modi ed. The next 50 years were dif cult for this theory because its experimental basis remained weak and its complicated mathematical structure was not well understood. However, in the subsequent period this theory ourished. Thanks to improvements in the te- nology and to the big progress in the methods of astronomical observations, the amount of observable facts to which general relativity is applicable was consid- ably enlarged. This is why general relativity is, today, one of the best experimentally tested theories while many competing theories could be disproved. Also the conc- tual and mathematical fundamentals are better understood now.

Gauge theory, symplectic geometry and symplectic topology are important areas at the crossroads of several mathematical disciplines. The present book, with expertly written surveys of recent developments in these areas, includes some of the first expository material of Seiberg-Witten theory, which has revolutionised the subjects since its introduction in late 1994. Topics covered include: introductions to Seiberg-Witten theory, to applications of the S-W theory to four-dimensional manifold topology, and to the classification of symplectic manifolds; an introduction to the theory of pseudo-holomorphic curves and to quantum cohomology; algebraically integrable Hamiltonian systems and moduli spaces; the stable topology of gauge theory, Morse-Floer theory; pseudo-convexity and its relations to symplectic geometry; generating functions; Frobenius manifolds and topological quantum field theory.

The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc. , give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.

Althoughsubmanifoldscomplexmanifoldshasbeenanactive?eldofstudyfor many years, in some sense this area is not su?ciently covered in the current literature. This text deals with the CR submanifolds of complex manifolds, with particular emphasis on CR submanifolds of complex projective space, and it covers the topics which are necessary for learning the basic properties of these manifolds. We are aware that it is impossible to give a complete overview of these submanifolds, but we hope that these notes can serve as an introduction to their study. We present the fundamental de?nitions and results necessary for reaching the frontiers of research in this ?eld. There are many monographs dealing with some current interesting topics in di?erential geometry, but most of these are written as encyclopedias, or research monographs, gathering recent results and giving the readers ample usefulinformationaboutthetopics. Therefore, thesekindsofmonographsare attractive to specialists in di?erential geometry and related ?elds and acce- able to professional di?erential geometers. However, for graduate students who are less advanced in di?erential geometry, these texts might be hard to read without assistance from their instructors. By contrast, the general philosophy of this book is to begin with the elementary facts about complex manifolds and their submanifolds, give some details and proofs, and introduce the reader to the study of CR submanifolds of complex manifolds; especially complex projective space. It includes only a few original results with precise proofs, while the others are cited in the reference list.

This book develops a modern presentation of Continuum Mechanics, oriented towards numerical applications in the ?elds of nonlinear analysis of solids, structures and ?uids. Kinematics of the continuum deformation, including pull-back/push-forward transformations between di erent con?gurations; stress and strain measures; objective stress rate and strain rate measures; balance principles; constitutive relations, with emphasis on elasto-plasticity of metals and variational prin- ples are developed using general curvilinear coordinates. Being tensor analysis the indispensable tool for the development of the continuum theory in general coordinates, in the appendix an overview of t- soranalysisisalsopresented. Embedded in the theoretical presentation, application examples are dev- oped to deepen the understanding of the discussed concepts. Even though the mathematical presentation of the di erent topics is quite rigorous; an e ort is made to link formal developments with engineering ph- ical intuition. This book is based on two graduate courses that the authors teach at the Engineering School of the University of Buenos Aires and it is intended for graduate engineering students majoring in mechanics and for researchers in the ?elds of applied mechanics and numerical methods. VIII Preface I am grateful to Klaus-Jurgen Bathe for introducing me to Computational Mechanics, for his enthusiasm, for his encouragement to undertake challenges and for his friendship."

The aim of this work is threefold: First it should be a monographical work on natural bundles and natural op erators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle A kT* M is in fact the value of a functor, which associates a bundle over M to each manifold M and a vector bundle homomorphism over f to each local diffeomorphism f between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that exterior derivative d transforms sections of A kT* M into sections of A k+1T* M for every manifold M can be expressed by saying that d is an operator from A kT* M into A k+1T* M."

curvilinear coordinates. This treatment includes in particular a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The fourth and last chapter, which heavily relies on Chapter 2, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are "two-dimensional," in the sense that they are expressed in terms of two curvilinear coordinates used for de?ning the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental "Korn inequality on a surface" and to an "in?nit- imal rigid displacement lemma on a surface." This chapter also includes a brief introduction to other two-dimensional shell equations. Interestingly, notions that pertain to di?erential geometry per se, suchas covariant derivatives of tensor ?elds, are also introduced in Chapters 3 and 4, where they appear most naturally in the derivation of the basic boundary value problems of three-dimensional elasticity and shell theory. Occasionally, portions of the material covered here are adapted from - cerpts from my book "Mathematical Elasticity, Volume III: Theory of Shells," published in 2000by North-Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to rely on such excerpts. Oth- wise, the bulk of this work was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].

The present volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such as closed curves and surfaces and other domain contours.

The first part of the book introduces the mathematical concept required for treating the manifolds considered. An introduction to the theory of motion of curves and surfaces is given. The second and third parts discuss the modeling of various physical solitons on compact systems.

This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces ("COMPASS"), which was held from September 29 to October 3, 2003, at Schloss Weinberg, Kefermarkt (A- tria). The workshop was mainly devoted to approximate algebraic geometry and its - plications. The organizers wanted to emphasize the novel idea of approximate implici- zation, that has strengthened the existing link between CAD / CAGD (Computer Aided Geometric Design) and classical algebraic geometry. The existing methods for exact implicitization (i. e., for conversion from the parametric to an implicit representation of a curve or surface) require exact arithmetic and are too slow and too expensive for industrial use. Thus the duality of an implicit representation and a parametric repres- tation is only used for low degree algebraic surfaces such as planes, spheres, cylinders, cones and toroidal surfaces. On the other hand, this duality is a very useful tool for - veloping ef?cient algorithms. Approximate implicitization makes this duality available for general curves and surfaces. The traditional exact implicitization of parametric surfaces produce global rep- sentations, which are exact everywhere. The surface patches used in CAD, however, are always de?ned within a small box only; they are obtained for a bounded parameter domain (typically a rectangle, or - in the case of "trimmed" surface patches - a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice."

From the reviews of the 1st edition:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph...

All the material, with few exceptions, is presented from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary.

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers."

Zentralblatt fur Mathematik 2004

For this 2nd edition, new results, new proofs, and commentaries for a better orientation of the reader have been added. In particular, in Chapter 7 several new sections concerning applications of the computer program "3-Manifold Recognizer" have been included. "

The present book is a translation and an expansion of lecture notes cor- sponding to a course of Mathematics of Control delivered during four years at the Ecole Nationale des Ponts et Chauss ees (Marne-la-Vall ee, France) to Master students. A reduced version of this course has also been given at the Master level at the University of Paris-Sud since eight years. It may the- fore serve as lecture notes for teaching at the Master or PhD level but also as a comprehensive introduction to researchers interested in atness and more generally in the mathematical theory of nite dimensional systems and c- trol. This book may be seen as an outcome of the applied research policy pi- oneered by the Ecole des Mines de Paris (now MINES-ParisTech), France, aiming not only at academic excellence, but also at collaborating with - dustries on speci c innovative projects to enhance technological innovation using the most advanced know-how. This in uence, though indirectly visible, mainly concerns the originality of some of the topics addressed here which are, in a sense, a theoretic synthesis of the author's applied contributions and viewpoints in the control eld, continuously elaborated and modi ed in contact with the industrial realities. Such a synthesis wouldn't have been made possible without the scienti c trust and nancial support of many c- panies during periods ranging from two to ten years.

This collection of survey lectures in mathematics traces the career of Beno Eckmann, whose work ranges across a broad spectrum of mathematical concepts from topology through homological algebra to group theory. One of our most influential living mathematicians, Eckmann has been associated for nearly his entire professional life with the Swiss Federal Technical University (ETH) at Zurich, as student, lecturer, professor, and professor emeritus.

The text of this book has its origins more than twenty- ve years ago. In the seminar of the Dutch Singularity Theory project in 1982 and 1983, the second-named author gave a series of lectures on Mixed Hodge Structures and Singularities, accompanied by a set of hand-written notes. The publication of these notes was prevented by a revolution in the subject due to Morihiko Saito: the introduction of the theory of Mixed Hodge Modules around 1985. Understanding this theory was at the same time of great importance and very hard, due to the fact that it uni es many di erent theories which are quite complicated themselves: algebraic D-modules and perverse sheaves. The present book intends to provide a comprehensive text about Mixed Hodge Theory with a view towards Mixed Hodge Modules. The approach to Hodge theory for singular spaces is due to Navarro and his collaborators, whose results provide stronger vanishing results than Deligne s original theory. Navarro and Guill en also lled a gap in the proof that the weight ltration on the nearby cohomology is the right one. In that sense the present book corrects and completes the second-named author s thesis."

This book is the first to systematically explore the classification and function theory of complex homogeneous bounded domains. The Siegel domains are discussed in detail, and proofs are presented. Using the normal Siegel domains to realize the homogeneous bounded domains, we can obtain more property of the geometry and the function theory on homogeneous bounded domains.

The purpose of these lecture notes is to provide an introduction to the theory of complex Monge-Ampere operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kahler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (Kahler-Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford-Taylor), Monge-Ampere foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi-Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kahler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli-Kohn-Nirenberg-Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong-Sturm and Berndtsson).

Each chapter can be read independently and is based on a series of lectures byR. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis."

This is the English translation of our introductory book on Teichmiiller space written in Japanese. We have taken advantage of the opportunity afforded by this translation to correct some minor errors in the original text, and to include several new related topics as additional sections or subsections. Among other things, we add the construction of Thurston's compactification of the Teichmiiller space in 4 of Chapter 3, and the Thurston and Bers theory on the classification of Teichmiiller modular transformations in 5 of Chapter 6. We also include a sketch of the proof of celebrated theorems of Royden ( 4 of Chapter 6), connection between Teichmiiller theory and deformation theory of the complex structures of Riemann surfaces due to Kodaira and Spencer ( 2.4 of Chapter 7), and a derivation of negativity of curvatures of the Teichmiiller space with respect to the Weil-Petersson metric ( 3.4 of Chapter 7). Further, we indicate how to verify that the compactified moduli space of a compact Riemann surface, constructed in Appendix B, is actually compact. We include several new references chiefly related to added parts. Other than these, the main body of the text is unchanged. Professors Tadashi Kuroda and Kotaro Oikawa kindly read the manuscript of the English edition and gave much valuable advice. We also record our gratitude to all friends and colleagues, especially Sadayoshi Kojima, Makoto Masumoto, Toshiyuki Sugawa, and Harumi Tanigawa, who rendered us much help by reading various parts of the manuscript."

This is the second volume of the new subseries "Invariant Theory and Algebraic Transformation Groups." The aim of the survey by A. Bialynicki-Birula is to present the main trends and achievements of research in the theory of quotients by actions of algebraic groups. This theory contains geometric invariant theory with various applications to problems of moduli theory. The contribution by J. Carrell treats the subject of torus actions on algebraic varieties, giving a detailed exposition of many of the cohomological results one obtains from having a torus action with fixed points. Many examples, such as toric varieties and flag varieties, are discussed in detail. W.M. McGovern studies the actions of a semisimple Lie or algebraic group on its Lie algebra via the adjoint action and on itself via conjugation. His contribution focuses primarily on nilpotent orbits that have found the widest application to representation theory in the last thirty-five years.

This volume is dedicated to the memory of Harry Ernest Rauch, who died suddenly on June 18, 1979. In organizing the volume we solicited: (i) articles summarizing Rauch's own work in differential geometry, complex analysis and theta functions (ii) articles which would give the reader an idea of the depth and breadth of Rauch's researches, interests, and influence, in the fields he investigated, and (iii) articles of high scientific quality which would be of general interest. In each of the areas to which Rauch made significant contribution - pinching theorems, teichmiiller theory, and theta functions as they apply to Riemann surfaces - there has been substantial progress. Our hope is that the volume conveys the originality of Rauch's own work, the continuing vitality of the fields he influenced, and the enduring respect for, and tribute to, him and his accom plishments in the mathematical community. Finally, it is a pleasure to thank the Department of Mathematics, of the Grad uate School of the City University of New York, for their logistical support, James Rauch who helped us with the biography, and Springer-Verlag for all their efforts in producing this volume. Isaac Chavel . Hershel M. Farkas Contents Harry Ernest Rauch - Biographical Sketch. . . . . . . . VII Bibliography of the Publications of H. E. Rauch. . . . . . X Ph.D. Theses Written under the Supervision of H. E. Rauch. XIII H. E. Rauch, Geometre Differentiel (by M. Berger) . . . . . . . ."

2 The authors of these issues involve not only mathematicians, but also speci alists in (mathematical) physics and computer sciences. So here the reader will find different points of view and approaches to the considered field. A. M. VINOGRADOV 3 Acta Applicandae Mathematicae 15: 3-21, 1989. (c) 1989 Kluwer Academic Publishers. Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results A. M. VINOORADOV Department of Mathematics, Moscow State University, 117234, Moscow, U. S. S. R. (Received: 22 August 1988) Abstract. The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed. AMS subject classifications (1980). 35A30, 58005, 58035, 58H05. Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions. o. Introduction In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations. Also, some general views and perspectives will be discussed.

Ever since its introduction around 1960 by Kirillov, the orbit method has played a major role in representation theory of Lie groups and Lie algebras. This book contains the proceedings of a conference held from August 29 to September 2, 1988, at the University of Copenhagen, about "the orbit method in representation theory." It contains ten articles, most of which are original research papers, by well-known mathematicians in the field, and it reflects the fact that the orbit method plays an important role in the representation theory of semisimple Lie groups, solvable Lie groups, and even more general Lie groups, and also in the theory of enveloping algebras.