The purpose of this monograph is to present the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Highlighted are the new precise results on the L(2) extension of holomorphic functions. In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L(2) method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka-Cartan theory is given by this method. The L(2) extension theorem with an optimal constant is included, obtained recently by Z. Blocki and by Q.-A. Guan and X.-Y. Zhou separately. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani-Yamaguchi, Berndtsson, and Guan-Zhou. Most of these results are obtained by the L(2) method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L(2) method obtained during these 15 years.

The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincare, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic.

In mathematical physics, the correspondence between quantum and classical mechanics is a central topic, which this book explores in more detail in the particular context of spin systems, that is, SU(2)-symmetric mechanical systems. A detailed presentation of quantum spin-j systems, with emphasis on the SO(3)-invariant decomposition of their operator algebras, is first followed by an introduction to the Poisson algebra of the classical spin system and then by a similarly detailed examination of its SO(3)-invariant decomposition. The book next proceeds with a detailed and systematic study of general quantum-classical symbol correspondences for spin-j systems and their induced twisted products of functions on the 2-sphere. This original systematic presentation culminates with the study of twisted products in the asymptotic limit of high spin numbers. In the context of spin systems it shows how classical mechanics may or may not emerge as an asymptotic limit of quantum mechanics. The book will be a valuable guide for researchers in this field and its self-contained approach also makes it a helpful resource for graduate students in mathematics and physics.

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

I.M.Gelfand, one of the leading contemporary mathematicians, largely determined the modern view of functional analysis with its numerous relations to other branches of mathematics, including mathematical physics, algebra, topology, differential geometry and analysis. With the publication of these Collected Papers in three volumes Gelfand gives a representative choice of his papers written in the last fifty years. Gelfand's research led to the development of remarkable mathematical theories - most now classics - in the field of Banach algebras, infinite-dimensional representations of Lie groups, the inverse Sturm-Liouville problem, cohomology of infinite-dimensional Lie algebras, integral geometry, generalized functions and general hypergeometric functions. The corresponding papers form the major part of the Collected Papers. Some articles on numerical methods and cybernetics as well as a few on biology are included. A substantial part of the papers have been translated into English especially for this edition. This edition is rounded off by a preface by S.G.Gindikin, a contribution by V.I.Arnold and an extensive bibliography with almost 500 references. Gelfand's Collected Papers will provide stimulating and serendipitous reading for researchers in a multitude of mathematical disciplines.

This textbook offers a high-level introduction to multi-variable differential calculus. Differential forms are introduced incrementally in the narrative, eventually leading to a unified treatment of Green's, Stokes' and Gauss' theorems. Furthermore, the presentation offers a natural route to differential geometry. Contents: Calculus of Vector Functions Tangent Spaces and 1-forms Line Integrals Differential Calculus of Mappings Applications of Differential Calculus Double and Triple Integrals Wedge Products and Exterior Derivatives Integration of Forms Stokes' Theorem and Applications

The book contains a survey of research on non-regular Riemannian geome try, carried out mainly by Soviet authors. The beginning of this direction oc curred in the works of A. D. Aleksandrov on the intrinsic geometry of convex surfaces. For an arbitrary surface F, as is known, all those concepts that can be defined and facts that can be established by measuring the lengths of curves on the surface relate to intrinsic geometry. In the case considered in differential is defined by specifying its first geometry the intrinsic geometry of a surface fundamental form. If the surface F is non-regular, then instead of this form it is convenient to use the metric PF' defined as follows. For arbitrary points X, Y E F, PF(X, Y) is the greatest lower bound of the lengths of curves on the surface F joining the points X and Y. Specification of the metric PF uniquely determines the lengths of curves on the surface, and hence its intrinsic geometry. According to what we have said, the main object of research then appears as a metric space such that any two points of it can be joined by a curve of finite length, and the distance between them is equal to the greatest lower bound of the lengths of such curves. Spaces satisfying this condition are called spaces with intrinsic metric. Next we introduce metric spaces with intrinsic metric satisfying in one form or another the condition that the curvature is bounded."

Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help applied mathematics, computer science and engineering students and researchers gain a firm mathematical grounding to use these tools confidently in their research. Its chart-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.

In these notes we consider two kinds of nonlinear evolution problems of von Karman type on Euclidean spaces of arbitrary even dimension. Each of these problems consists of a system that results from the coupling of two highly nonlinear partial differential equations, one hyperbolic or parabolic and the other elliptic. These systems take their name from a formal analogy with the von Karman equations in the theory of elasticity in two dimensional space. We establish local (respectively global) results for strong (resp., weak) solutions of these problems and corresponding well-posedness results in the Hadamard sense. Results are found by obtaining regularity estimates on solutions which are limits of a suitable Galerkin approximation scheme. The book is intended as a pedagogical introduction to a number of meaningful application of classical methods in nonlinear Partial Differential Equations of Evolution. The material is self-contained and most proofs are given in full detail. The interested reader will gain a deeper insight into the power of nontrivial a priori estimate methods in the qualitative study of nonlinear differential equations.

Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. Today it is possible to describe differential geometry as "the study of structures on the tangent space," and this text develops this point of view. This book, unlike other introductory texts in differential geometry, develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. The main goal of this book is to bring the undergraduate student who already has a solid foundation in the standard mathematics curriculum into contact with the beauty of higher mathematics. In particular, the presentation here emphasizes the consequences of a definition and the careful use of examples and constructions in order to explore those consequences.

During his short life Oswald Teichmuller wrote 34 papers, all reproduced in this volume. From the Preface: "Teichmuller's most influential paper was called "Extremale quasikonforme Abbildungen und quadratische Differentiale" (No. 20 in this collection). At the time of its appearance several special cases of extremal problems for quasiconformal mappings had already been solved, and Teichmuller was able to draw on a substantial fund of experience. Nevertheless, it was a remarkable feat to extract the common features of all the known examples and formulate a conjecture, now known as Teichmuller's theorem, which in an unexpected way connects the holomorphic second order differentials on a Riemann surface with the extremal quasiconformal mappings of that surface. The paper of 1939 contains a uniqueness proof, which is essentially still the only known proof, but not yet a rigorous existence proof. This did not prevent Teichmuller from laying the foundation of what has become known as the theory of Teichmuller spaces, a theory that has mushroomed to an extent that could not then have been foreseen. At the same time Teichmuller's work led to a deeper understanding of the fundamental role played by quasiconformal mappings in all of geometric function theory, and it foreshadowed the subsequent development of the theory of quasiconformal mappings in several dimensions...the whole theory of analytic functions of one complex variable has been greatly enriched by the inclusion of quasiconformal mappings, much of it based on Teichmuller's seminal ideas."

In recent years, research in K3 surfaces and Calabi-Yau varieties has seen spectacular progress from both arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics-in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds, held at the Fields Institute (August 16-25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi-Yau varieties. With the big variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated. Unlike most other conferences, the 2011 Calabi-Yau workshop started with 3 days of introductory lectures. A selection of 4 of these lectures is included in this volume. These lectures can be used as a starting point for the graduate students and other junior researchers, or as a guide to the subject.

The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are "minimal" in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

This book illustrates the deep roots of the geometrically nonlinear kinematics of generalized continuum mechanics in differential geometry. Besides applications to first- order elasticity and elasto-plasticity an appreciation thereof is particularly illuminating for generalized models of continuum mechanics such as second-order (gradient-type) elasticity and elasto-plasticity. After a motivation that arises from considering geometrically linear first- and second- order crystal plasticity in Part I several concepts from differential geometry, relevant for what follows, such as connection, parallel transport, torsion, curvature, and metric for holonomic and anholonomic coordinate transformations are reiterated in Part II. Then, in Part III, the kinematics of geometrically nonlinear continuum mechanics are considered. There various concepts of differential geometry, in particular aspects related to compatibility, are generically applied to the kinematics of first- and second- order geometrically nonlinear continuum mechanics. Together with the discussion on the integrability conditions for the distortions and double-distortions, the concepts of dislocation, disclination and point-defect density tensors are introduced. For concreteness, after touching on nonlinear fir st- and second-order elasticity, a detailed discussion of the kinematics of (multiplicative) first- and second-order elasto-plasticity is given. The discussion naturally culminates in a comprehensive set of different types of dislocation, disclination and point-defect density tensors. It is argued, that these can potentially be used to model densities of geometrically necessary defects and the accompanying hardening in crystalline materials. Eventually Part IV summarizes the above findings on integrability whereby distinction is made between the straightforward conditions for the distortion and the double-distortion being integrable and the more involved conditions for the strain (metric) and the double-strain (connection) being integrable. The book addresses readers with an interest in continuum modelling of solids from engineering and the sciences alike, whereby a sound knowledge of tensor calculus and continuum mechanics is required as a prerequisite.

This is the second edition of this best selling problem book for students, now containing over 400 completely solved exercises on differentiable manifolds, Lie theory, fibre bundles and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. Many of the definitions and theorems used throughout are explained in the first section of each chapter where they appear. A 56-page collection of formulae is included which can be useful as an aide-memoire, even for teachers and researchers on those topics. In this 2nd edition: * 76 new problems * a section devoted to a generalization of Gauss' Lemma * a short novel section dealing with some properties of the energy of Hopf vector fields * an expanded collection of formulae and tables * an extended bibliography Audience This book will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics and some branches of engineering with a rudimentary knowledge of linear and multilinear algebra.

Starting from an undergraduate level, this book systematically develops the basics of * Calculus on manifolds, vector bundles, vector fields and differential forms, * Lie groups and Lie group actions, * Linear symplectic algebra and symplectic geometry, * Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.

Traditionally, Lorentzian geometry has been used as a necessary tool to understand general relativity, as well as to explore new genuine geometric behaviors, far from classical Riemannian techniques. Recent progress has attracted a renewed interest in this theory for many researchers: long-standing global open problems have been solved, outstanding Lorentzian spaces and groups have been classified, new applications to mathematical relativity and high energy physics have been found, and further connections with other geometries have been developed. Samples of these fresh trends are presented in this volume, based on contributions from the VI International Meeting on Lorentzian Geometry, held at the University of Granada, Spain, in September, 2011. Topics such as geodesics, maximal, trapped and constant mean curvature submanifolds, classifications of manifolds with relevant symmetries, relations between Lorentzian and Finslerian geometries, and applications to mathematical physics are included. This book will be suitable for a broad audience of differential geometers, mathematical physicists and relativists, and researchers in the field.

The study of qualitative aspects of PDE's has always attracted much attention from the early beginnings. More recently, once basic issues about PDE's, such as existence, uniqueness and stability of solutions, have been understood quite well, research on topological and/or geometric properties of their solutions has become more intense. The study of these issues is attracting the interest of an increasing number of researchers and is now a broad and well-established research area, with contributions that often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name a few. This volume collects a selection of original results and informative surveys by a group of international specialists in the field, analyzes new trends and techniques and aims at promoting scientific collaboration and stimulating future developments and perspectives in this very active area of research.

The mathematical theory of "open" dynamical systems is a creation of the twentieth century. Its humble beginnings focused on ideas of Laplace transforms applied to linear problems of automatic control and to the analysis and synthesis of electrical circuits. However during the second half of the century, it flowered into a field based on an array of sophisticated mathematical concepts and techniques from algebra, nonlinear analysis and differential geometry. The central notion is that of a dynamical system that exchanges matter, energy, or information with its surroundings, i.e. an "open" dynamical system. The mathema tization of this notion evolved considerably over the years. The early development centered around the input/output point of view and led to important results, particularly in controller design. Thinking about open systems as a "black box" that accepts stimuli and produces responses has had a wide influence also in areas outside engineering, for example in biology, psychology, and economics. In the early 1960's, especially through the work of Kalman, input/state/output models came in vogue. This model class accommodates very nicely the internal initial conditions that are essentially always present in a dynamical system. The introduction of input/state/output models led to a tempestuous development that made systems and control into a mature discipline with a wide range of concepts, results, algorithms, and applications.

This book is based on the experience of teaching the subject by the author in Russia, France, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics on tensors, Riemannian geometry and geometric approach to partial differential equations. Application of approximate transformation groups to the equations of general relativity in the de Sitter space simplifies the subject significantly.

Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit of Kashiwara and Vergne's original work for Lie groups. The book includes a complete rewriting of several articles by the author, updated and improved following Alekseev, Meinrenken and Torossian's recent proofs of the conjecture. The chapters are largely independent of each other. Some open problems are suggested to encourage future research. It is aimed at graduate students and researchers with a basic knowledge of Lie theory.

As many readers will know, the 20th century was a time when the fields of mathematics and the sciences were seen as two separate entities. Caused by the rapid growth of the physical sciences and an increasing abstraction in mathematical research, each party, physicists and mathematicians alike, suffered a misconception; not only of the opposition's theoretical underpinning, but of how the two subjects could be intertwined and effectively utilized. One sub-discipline that played a part in the union of the two subjects is Theoretical Physics. Breaking it down further came the fundamental theories, Relativity and Quantum theory, and later on Yang-Mills theory. Other areas to emerge in this area are those derived from the works of Donaldson, Chern-Simons, Floer-Fukaya, and Seiberg-Witten. Aimed at a wide audience, Physical Topics in Mathematics demonstrates how various physical theories have played a crucial role in the developments of Mathematics and in particular, Geometric Topology. Issues are studied in great detail, and the book steadfastly covers the background of both Mathematics and Theoretical Physics in an effort to bring the reader to a deeper understanding of their interaction. Whilst the world of Theoretical Physics and Mathematics is boundless; it is not the intention of this book to cover its enormity. Instead, it seeks to lead the reader through the world of Physical Mathematics; leaving them with a choice of which realm they wish to visit next.

In this text, integral geometry deals with Radon's problem of representing a function on a manifold in terms of its integrals over certain submanifolds-hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial di erential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: "Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. the contents of the book is concentrated around the duality and double vibration, which is realized through the masterful treatment of a variety of examples. the book is written by an expert, who has made fundamental contributions to the area." -Boris Rubin, Louisiana State University

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the "tropical" approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as "degenerations" of the corresponding algebro-geometric objects.

Based on a two-semester course aimed at illustrating various interactions of "pure mathematics" with other sciences, such as hydrodynamics, thermodynamics, statistical physics and information theory, this text unifies three general topics of analysis and physics, which are as follows: the dimensional analysis of physical quantities, which contains various applications including Kolmogorov's model for turbulence; functions of very large number of variables and the principle of concentration along with the non-linear law of large numbers, the geometric meaning of the Gauss and Maxwell distributions, and the Kotelnikov-Shannon theorem; and, finally, classical thermodynamics and contact geometry, which covers two main principles of thermodynamics in the language of differential forms, contact distributions, the Frobenius theorem and the Carnot-Caratheodory metric. It includes problems, historical remarks, and Zorich's popular article, "Mathematics as language and method."