The five lectures presented in this volume address very timely mathematical problems in relativity and cosmology. "Part I" is devoted to the initial value and evolution problems of the Einstein equations. Especially it deals with the Einstein-Yang-Mills-Boltzmann system, fluid models with finite or infinite conductivity, global evolution of a new (two-phase) model for gravitational collapse and the structure of maximal, asymptotically flat, vacuum solutions of the constraint equations which have the additional property of containing trapped surfaces. "Part II" focuses on geometrical-topological problems in relativity and cosmology: on the role of cosmic censorship for the global structure of the Einstein-Maxwell equations and on the mathematical structure of quantum conformal superspace.

In recognition of professor Shiing-Shen Chern's long and distinguished service to mathematics and to the University of California, the geometers at Berkeley held an International Symposium in Global Analysis and Global Geometry in his honor in June 1979. The output of this Symposium was published in a series of three separate volumes, comprising approximately a third of Professor Chern's total publications up to 1979. Later, this fourth volume was published, focusing on papers written during the Eighties.

This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold. The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications. Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part. The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis. The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.

Our purpose and main concern in writing this book is to illuminate classical concepts from the noncommutative viewpoint, to make the language and techniques of noncommutative geometry accessible and familiar to practi- tioners of classical mathematics, and to benefit physicists interested in the uses of noncommutative spaces. Same may say that ours is a very "com- mutative" way to deal with noncommutative matters; this charge we readily admit. Noncommutative geometry amounts to a program of unification of math- ematics under the aegis of the quantum apparatus, i.e., the theory of ope- rators and of C*-algebras. Largely the creation of a single person, Alain Connes, noncommutative geometry is just coming of age as the new century opens. The bible of the subject is, and will remain, Connes' Noncommuta- tive Geometry (1994), itself the "3.8-fold expansion" of the French Geome- trie non commutative ( 1990). Theseare extraordinary books, a "tapestry" of physics and mathematics, in the words of Vaughan jones, and the work of a "poet of modern science," according to Daniel Kastler, replete with subtle knowledge and insights apt to inspire several generations.

This treatment of differential geometry and the mathematics required for general relativity makes the subject accessible, for the first time, to anyone familiar with elementary calculus in one variable and with some knowledge of vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as the book form will allow.

This monograph contains a description of original methods and results concern ing global properties of linear differential equations of the nth order, n ~ 2, in the real domain. This area of research concerning second order linear differential equations was started 35 years ago by O. Boruvka. He summarized his results in the monograph "Lineare Differentialtransforrnationen 2. Ordnung", VEB, Berlin 1967 (extended version: "Linear Differential Transformations of the Second Order", The English U niv. Press, London 1971). This book deals with linear differential equations of the nth order, n ~ 2, and summarizes results in this field in a unified fashion. However, this mono graph is by no means intended to be a survey of all results in this area. I t contains only a selection of results, which serves to illustrate the unified approach presented here. By using recent methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and linear differential equations of the second order, and by introducing several original methods, global solutions of problems which were previously studied only locally by Kummer, Brioschi, Laguerre, Forsyth, Halphen, Lie, Stiickel and others are provided. The structure of global transformations is described by algebraic means (theory of categories: Brandt and Ehresmann groupoids), a new geometrical approach is introduced that leads to global canonical forms (in contrast to the local Laguerre-Forsyth or Halphen forms) and is suitable for the application of Cartan's moving-frame-of-reference method.

In recognition of professor Shiing-Shen Chern s long and distinguished service to mathematics and to the University of California, the geometers at Berkeley held an International Symposium in Global Analysis and Global Geometry in his honor in June 1979. The output of this Symposium was published in a series of three separate volumes, comprising approximately a third of Professor Chern s total publications up to 1979. Later, a fourth volume was published, focusing on papers written during the Eighties. This second volume comprises selected papers written between 1932 and 1965.

This volume contains the Proceedings of the conference "Complex and Differential Geometry 2009", held at Leibniz Universitat Hannover, September 14 - 18, 2009. It was the aim of this conference to bring specialists from differential geometry and (complex) algebraic geometry together and to discuss new developments in and the interaction between these fields. Correspondingly, the articles in this book cover a wide area of topics, ranging from topics in (classical) algebraic geometry through complex geometry, including (holomorphic) symplectic and poisson geometry, to differential geometry (with an emphasis on curvature flows) and topology.

Noncommutative differential geometry is a new approach to classical geometry. It was originally used by Fields Medalist A. Connes in the theory of foliations, where it led to striking extensions of Atiyah-Singer index theory. It also may be applicable to hitherto unsolved geometric phenomena and physical experiments. However, noncommutative differential geometry was not well understood even among mathematicians. Therefore, an international symposium on commutative differential geometry and its applications to physics was held in Japan, in July 1999. Topics covered included: deformation problems, Poisson groupoids, operad theory, quantization problems, and D-branes. The meeting was attended by both mathematicians and physicists, which resulted in interesting discussions. This volume contains the refereed proceedings of this symposium. Providing a state of the art overview of research in these topics, this book is suitable as a source book for a seminar in noncommutative geometry and physics.

The aim of this book is to provide a short but complete exposition of the logical structure of classical relativistic electrodynamics written in the language and spirit of coordinate-free differential geometry. The intended audience is primarily mathematicians who want a bare-bones account of the foundations of electrodynamics written in language with which they are familiar and secondarily physicists who may be curious how their old friend looks in the new clothes of the differential-geometric viewpoint which in recent years has become an important language and tool for theoretical physics. This work is not intended to be a textbook in electrodynamics in the usual sense; in particular no applications are treated, and the focus is exclusively the equations of motion of charged particles. Rather, it is hoped that it may serve as a bridge between mathemat ics and physics. Many non-physicists are surprised to learn that the correct equation to describe the motion of a classical charged particle is still a matter of some controversy. The most mentioned candidate is the Lorentz-Dirac equation t . However, it is experimentally unverified, is known to have no physically reasonable solutions in certain circumstances, and its usual derivations raise serious foundational issues. Such difficulties are not extensively discussed in most electrodynamics texts, which quite naturally are oriented toward applying the well-verified part of the subject to con crete problems."

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit ful again. Many of the leading specialists of both fields became involved in this devel opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, *we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth while.

Addressing graduate students and researchers in physics and mathematics, this book fills a gap in the literature. It is an introduction into modern constructive physics, field theory and statistical mechanics and a survey on the most recent research in this field. It presents the main technical tools such as cluster expansion and their implementation in the rigorous renormalization group, and studies physical models in some detail. The reader will find a study of the ultraviolet limit of the Gross-Neveu model, of continuous symmetry breaking and of self-avoiding random walks in statistical mechanics, as well as applications to solid-state physics. Mathematicians will find constructive methods useful for studies in partial differential equations.

This volume presents a complete and self-contained description of new results in the theory of manifolds of nonpositive curvature. It is based on lectures delivered by M. Gromov at the College de France in Paris. Therefore this book may also serve as an introduction to the subject of nonpositively curved manifolds. The latest progress in this area is reflected in the article of W. Ballmann describing the structure of manifolds of higher rank.

The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and remains a useful and powerful tool in such areas as Lie theory, representation theory, integrable systems, complex geometry, and mathematical physics. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits (1962), places him as the founder of orbit theory. The original research papers in this volume are written by prominent mathematicians and reflect recent achievements in orbit theory and other closely related areas such as harmonic analysis, classical representation theory, Lie superalgebras, Poisson geometry, and quantization. Contributors: A. Alekseev, J. Alev, V. Baranovksy, R. Brylinski, J. Dixmier, S. Evens, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, P.W. Michor, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses.

This English edition could serve as a text for a first year graduate course on differential geometry, as did for a long time the Chicago Notes of Chern mentioned in the Preface to the German Edition. Suitable references for ordin ary differential equations are Hurewicz, W. Lectures on ordinary differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, New York, 1977. Upon David Hoffman fell the difficult task of transforming the tightly constructed German text into one which would mesh well with the more relaxed format of the Graduate Texts in Mathematics series. There are some e1aborations and several new figures have been added. I trust that the merits of the German edition have survived whereas at the same time the efforts of David helped to elucidate the general conception of the Course where we tried to put Geometry before Formalism without giving up mathematical rigour. 1 wish to thank David for his work and his enthusiasm during the whole period of our collaboration. At the same time I would like to commend the editors of Springer-Verlag for their patience and good advice. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German Edition This book has its origins in a one-semester course in differential geometry which 1 have given many times at Gottingen, Mainz, and Bonn."

The present volume grew out of the Heidelberg Knot Theory Semester, organized by the editors in winter 2008/09 at Heidelberg University. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical physics and cell biology. Both original research and survey articles are presented; numerous illustrations support the text. The book will be of great interest to researchers in topology, geometry, and mathematical physics, graduate students specializing in knot theory, and cell biologists interested in the topology of DNA strands.

Several well-established geometric and topological methods are used in this work in an application to a beautiful physical phenomenon known as the geometric phase. This book examines the geometric phase, bringing together different physical phenomena under a unified mathematical scheme. The material is presented so that graduate students and researchers in applied mathematics and physics with an understanding of classical and quantum mechanics can handle the text.

This edited survey book consists of 20 chapters showing application of Clifford algebra in quantum mechanics, field theory, spinor calculations, projective geometry, Hypercomplex algebra, function theory and crystallography. Many examples of computations performed with a variety of readily available software programs are presented in detail.

This book provides definitions and mathematical derivations of fundamental relationships of tensor analysis encountered in nonlinear continuum mechanics and continuum physics, with a focus on finite deformation kinematics and classical differential geometry. Of particular interest are anholonomic aspects arising from a multiplicative decomposition of the deformation gradient into two terms, neither of which in isolation necessarily obeys the integrability conditions satisfied by the gradient of a smooth vector field. The concise format emphasizes clarity and ease of reference, and detailed step-by-step derivations of most analytical results are provided.

This is a monograph on geometrical and topological features which arise in various quantization procedures. Quantization schemes consider the feasibility of arriving at a quantum system from a classical one and these involve three major procedures viz. i) geometric quantization, ii) Klauder quantization, and iii) stochastic quanti zation. In geometric quantization we have to incorporate a hermitian line bundle to effectively generate the quantum Hamiltonian operator from a classical Hamil tonian. Klauder quantization also takes into account the role of the connection one-form along with coordinate independence. In stochastic quantization as pro posed by Nelson, Schrodinger equation is derived from Brownian motion processes; however, we have difficulty in its relativistic generalization. It has been pointed out by several authors that this may be circumvented by formulating a new geometry where Brownian motion proceses are considered in external as well as in internal space and, when the complexified space-time is considered, the usual path integral formulation is achieved. When this internal space variable is considered as a direc tion vector introducing an anisotropy in the internal space, we have the quantization of a Fermi field. This helps us to formulate a stochastic phase space formalism when the internal extension can be treated as a gauge theoretic extension. This suggests that massive fermions may be considered as Skyrme solitons. The nonrelativistic quantum mechanics is achieved in the sharp point limit."

The implicit function theorem is part ofthe bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis.

There are many different forms of the implicit function theorem, including (i) the classical formulation for"Ck"functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, and (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present uncorrected reprint of this classicmonograph.

Originally published in 2002, "The Implicit Function Theorem"is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and placein context a substantial body of mathematical ideas. "

Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann.

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateaus problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateaus problem have no interior branch points.

The main subject of the book is the full understanding of Weyl's formula for the volume of a tube, its roots and its implications. Another discussed approach to the study of volumes of tubes is the computation of the power series of the volume of a tube as a function of its radius. The chapter on mean values, besides its intrinsic interest, shows an interesting fact: methods which are useful for volumes are also useful for problems related with the Laplacian. Historical notes and Mathematica drawings have been added to this revised second edition.

Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.