Edited in collaboration with the Grassmann Research Group, this book contains many important articles delivered at the ICM 2014 Satellite Conference and the 18th International Workshop on Real and Complex Submanifolds, which was held at the National Institute for Mathematical Sciences, Daejeon, Republic of Korea, August 10-12, 2014. The book covers various aspects of differential geometry focused on submanifolds, symmetric spaces, Riemannian and Lorentzian manifolds, and Kahler and Grassmann manifolds.

The literature on natural bundles and natural operators in differential geometry, was until now, scattered in the mathematical journal literature. This book is the first monograph on the subject, collecting this material in a unified presentation. The book begins with an introduction to differential geometry stressing naturality and functionality, and the general theory of connections on arbitrary fibered manifolds. The functional approach to classical natural bundles is extended to a large class of geometrically interesting categories. Several methods of finding all natural operators are given and these are identified for many concrete geometric problems. After reduction each problem to a finite order setting, the remaining discussion is based on properties of jet spaces, and the basic structures from the theory of jets are therefore described here too in a self-contained manner. The relations of these geometric problems to corresponding questions in mathematical physics are brought out in several places in the book, and it closes with a very comprehensive bibliography of over 300 items. This book is a timely addition to literature filling the gap that existed here and will be a standard reference on natural operators for the next few years.

Singularity theory stands at a cross-road of mathematics, a meeting point where manyareasofmathematicscometogether,suchasgeometry,topologyandalgebra, analysis,di?erential equations and dynamical systems, combinatoricsand number theory, to mention some of them. Thus, one who would write a book about this fascinatingtopicnecessarilyfacesthechallengeofhavingtochoosewhattoinclude and,mostdi?cult,whatnottoinclude. Acomprehensivetreatmentofsingularities would have to consist of a collection of books, which would be beyond our present scope. Hence this work does not pretend to be comprehensive of the subject, neither is it a text book with a systematic approachto singularitytheory asa core idea. Thisisrather a collectionof essaysonselected topicsaboutthe topologyand geometry of real and complex analytic spaces around their isolated singularities. I have worked in the area of singularities since the late 1970s, and during this time have had the good fortune of encountering many gems of mathematics concerningthetopologyofsingularitiesandrelatedtopics,masterpiecescreatedby greatmathematicians like Riemann, Klein and Poincar' e,then Milnor, Hirzebruch, Thom, Mumford, Brieskorn, Atiyah, Arnold, Wall, LeDung " Tran ' g, Neumann, Looijenga, Teissier, and many more whose names I cannot include since the list would be too long and, even that, I would leave aside important names. My own research has always stood on the shoulders of all of them. In taking this broad approach I realize how di?cult it is to present an overall picture of the myriad of outstanding contributions in this area of mathematics during the last century, since they are scattered in very many books and research articles.

To facilitate a deeper understanding of tensegrity structures, this book focuses on their two key design problems: self-equilibrium analysis and stability investigation. In particular, high symmetry properties of the structures are extensively utilized. Conditions for self-equilibrium as well as super-stability of tensegrity structures are presented in detail. An analytical method and an efficient numerical method are given for self-equilibrium analysis of tensegrity structures: the analytical method deals with symmetric structures and the numerical method guarantees super-stability. Utilizing group representation theory, the text further provides analytical super-stability conditions for the structures that are of dihedral as well as tetrahedral symmetry. This book not only serves as a reference for engineers and scientists but is also a useful source for upper-level undergraduate and graduate students. Keeping this objective in mind, the presentation of the book is self-contained and detailed, with an abundance of figures and examples.

This book presents a comprehensive, encyclopedic approach to the subject of foliations, one of the major concepts of modern geometry and topology. It addresses graduate students and researchers and serves as a reference book for experts in the field.

The International Conference on Finsler and Lagrange Geometry and its Applications: A Meeting of Minds, took place August 13-20, 1998 at the University of Alberta in Edmonton, Canada. The main objective of this meeting was to help acquaint North American geometers with the extensive modern literature on Finsler geometry and Lagrange geometry of the Japanese and European schools, each with its own venerable history, on the one hand, and to communicate recent advances in stochastic theory and Hodge theory for Finsler manifolds by the younger North American school, on the other. The intent was to bring together practitioners of these schools of thought in a Canadian venue where there would be ample opportunity to exchange information and have cordial personal interactions. The present set of refereed papers begins .with the Pedagogical Sec tion I, where introductory and brief survey articles are presented, one from the Japanese School and two from the European School (Romania and Hungary). These have been prepared for non-experts with the intent of explaining basic points of view. The Section III is the main body of work. It is arranged in alphabetical order, by author. Section II gives a brief account of each of these contribu tions with a short reference list at the end. More extensive references are given in the individual articles."

This book is an outgrowth of the Workshop on "Regulators in Analysis, Geom etry and Number Theory" held at the Edmund Landau Center for Research in Mathematical Analysis of The Hebrew University of Jerusalem in 1996. During the preparation and the holding of the workshop we were greatly helped by the director of the Landau Center: Lior Tsafriri during the time of the planning of the conference, and Hershel Farkas during the meeting itself. Organizing and running this workshop was a true pleasure, thanks to the expert technical help provided by the Landau Center in general, and by its secretary Simcha Kojman in particular. We would like to express our hearty thanks to all of them. However, the articles assembled in the present volume do not represent the proceedings of this workshop; neither could all contributors to the book make it to the meeting, nor do the contributions herein necessarily reflect talks given in Jerusalem. In the introduction, we outline our view of the theory to which this volume intends to contribute. The crucial objective of the present volume is to bring together concepts, methods, and results from analysis, differential as well as algebraic geometry, and number theory in order to work towards a deeper and more comprehensive understanding of regulators and secondary invariants. Our thanks go to all the participants of the workshop and authors of this volume. May the readers of this book enjoy and profit from the combination of mathematical ideas here documented.

This is the Proceedings of the ICM 2010 Satellite Conference on "Buildings, Finite Geometries and Groups" organized at the Indian Statistical Institute, Bangalore, during August 29 - 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups.

The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

The erratic motion of pollen grains and other tiny particles suspended in liquid is known as Brownian motion, after its discoverer, Robert Brown, a botanist who worked in 1828, in London. He turned over the problem of why this motion occurred to physicists who were investigating kinetic theory and thermodynamics; at a time when the existence of molecules had yet to be established. In 1900, Henri Poincare lectured on this topic to the 1900 International Congress of Physicists, in Paris [Wic95]. At this time, Louis Bachelier, a thesis student of Poincare, made a monumental breakthrough with his Theory of Stock Market Fluctuations, which is still studied today, [Co064]. Norbert Wiener (1923), who was first to formulate a rigorous concept of the Brownian path, is most often cited by mathematicians as the father of the subject, while physicists will cite A. Einstein (1905) and M. Smoluchowski. Both considered Markov diffusions and realized that Brownian behaviour nd could be formulated in terms of parabolic 2 order linear p. d. e. 'so Further more, from this perspective, the covariance of changes in position could be allowed to depend on the position itself, according to the invariant form of the diffusion introduced by Kolmogorov in 1937, [KoI37]. Thus, any time homogeneous Markov diffusion could be written in terms of the Laplacian, intrinsically given by the symbol (covariance) of the p. d. e. , plus a drift vec tor. The theory was further advanced in 1949, when K.

This is the first exposition of the quantization theory of singular symplectic (Marsden-Weinstein) quotients and their applications to physics. The reader will acquire an introduction to the various techniques used in this area, as well as an overview of the latest research approaches. These involve classical differential and algebraic geometry, as well as operator algebras and noncommutative geometry. Thus one will be amply prepared to follow future developments in this field.

This introduction to the theory of Diophantine approximation pays special regard to Schmidt's subspace theorem and to its applications to Diophantine equations and related topics. The geometric viewpoint on Diophantine equations has been adopted throughout the book. It includes a number of results, some published here for the first time in book form, and some new, as well as classical material presented in an accessible way. Graduate students and experts alike will find the book's broad approach useful for their work, and will discover new techniques and open questions to guide their research. It contains concrete examples and many exercises (ranging from the relatively simple to the much more complex), making it ideal for self-study and enabling readers to quickly grasp the essential concepts.

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," etale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

From a historical point of view, the theory we submit to the present study has its origins in the famous dissertation of P. Finsler from 1918 ([Fi]). In a the classical notion also conventional classification, Finsler geometry has besides a number of generalizations, which use the same work technique and which can be considered self-geometries: Lagrange and Hamilton spaces. Finsler geometry had a period of incubation long enough, so that few math ematicians (E. Cartan, L. Berwald, S.S. Chem, H. Rund) had the patience to penetrate into a universe of tensors, which made them compare it to a jungle. To aU of us, who study nowadays Finsler geometry, it is obvious that the qualitative leap was made in the 1970's by the crystallization of the nonlinear connection notion (a notion which is almost as old as Finsler space, [SZ4]) and by work-skills into its adapted frame fields. The results obtained by M. Matsumoto (coUected later, in 1986, in a monograph, [Ma3]) aroused interest not only in Japan, but also in other countries such as Romania, Hungary, Canada and the USA, where schools of Finsler geometry are founded and are presently widely recognized.

A fascinating tour through parts of geometry students are unlikely to see in the rest of their studies while, at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclids fifth postulate lead to interesting and different patterns and symmetries, and, in the process of examining geometric objects, the author incorporates the algebra of complex and hypercomplex numbers, some graph theory, and some topology. Interesting problems are scattered throughout the text. Nevertheless, the book merely assumes a course in Euclidean geometry at high school level. While many concepts introduced are advanced, the mathematical techniques are not. Singers lively exposition and off-beat approach will greatly appeal both to students and mathematicians, and the contents of the book can be covered in a one-semester course, perhaps as a sequel to a Euclidean geometry course.

M-Solid Varieties of Algebras provides a complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on M-solid varieties of semirings and semigroups. The book aims to develop the theory of M-solid varieties as a system of mathematical discourse that is applicable in several concrete situations. It applies the general theory to two classes of algebraic structures, semigroups and semirings. Both these varieties and their subvarieties play an important role in computer science.

A unique feature of this book is the use of Galois connections to integrate different topics. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories. This concept is used throughout the whole book, along with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators.

This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.

After some decades of work a satisfactory theory of quantum gravity is still not available; moreover, there are indications that the original field theoretical approach may be better suited than originally expected. There, to first approximation, one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes. Ingredients of this approach are the formulation of quantum physics in terms of C*-algebras, the geometry of Lorentzian manifolds, in particular their causal structure, and linear hyperbolic differential equations where the well-posedness of the Cauchy problem plays a distinguished role, as well as more recently the insights from suitable concepts such as microlocal analysis. This primer is an outgrowth of a compact course given by the editors and contributing authors to an audience of advanced graduate students and young researchers in the field, and assumes working knowledge of differential geometry and functional analysis on the part of the reader.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

After a brief description of the evolution of thinking on Finslerian geometry starting from Riemann, Finsler, Berwald and Elie Cartan, the book gives a clear and precise treatment of this geometry. The first three chapters develop the basic notions and methods, introduced by the author, to reach the global problems in Finslerian Geometry. The next five chapters are independent of each other, and deal with among others the geometry of generalized Einstein manifolds, the classification of Finslerian manifolds of constant sectional curvatures. They also give a treatment of isometric, affine, projective and conformal vector fields on the unitary tangent fibre bundle.

Key features

- Theory of connections of vectors and directions on the unitary tangent fibre bundle.
- Complete list of Bianchi identities for a regular conection of directions.
- Geometry of generalized Einstein manifolds.
- Classification of Finslerian manifolds.
- Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle.
- Theory of connections of vectors and directions on the unitary tangent fibre bundle.
- Complete list of Bianchi identities for a regular conection of directions.
- Geometry of generalized Einstein manifolds.
- Classification of Finslerian manifolds.
- Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle.

Differential Manifold is the framework of particle physics and astrophysics nowadays. It is important for all research physicists to be well accustomed to it and even experimental physicists should be able to manipulate equations and expressions in that framework.This book gives a comprehensive description of the basics of differential manifold with a full proof of any element. A large part of the book is devoted to the basic mathematical concepts in which all necessary for the development of the differential manifold is expounded and fully proved.This book is self-consistent: it starts from first principles. The mathematical framework is the set theory with its axioms and its formal logic. No special knowledge is needed.

This book is a compilation of all basic topics of Analytical Geometry of Two Dimensions and is intended to serve as an introductory text aimed towards undergraduate and graduate students in science and technology. An understanding of basic school level algebra and geometry can serve as the prerequisite for following this book. The present work is no original work but an attempt to make the subject thoroughly intelligible. All the important properties of the conics have been discussed either in the articles or in illustrative examples. Each chapter has sufficient completely solved problems and a set of carefully graded and motivating unsolved exercises. Please note: Taylor & Francis does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.

The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that "in learning the sciences examples are of more use than precepts". We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a "global and analytical bias". We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincare duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hoelder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.

Minkowski geometry is a type of non-Euclidean geometry in a finite number of dimensions in which distance is not 'uniform' in all directions. This book presents the first comprehensive treatment of Minkowski geometry since the 1940s. The author begins by describing the fundamental metric properties and the topological properties of existence of Minkowski space. This is followed by a treatment of two-dimensional spaces and characterisations of Euclidean space among normed spaces. The central three chapters present the theory of area and volume in normed spaces, a fascinating geometrical interplay among the various roles of the ball in Euclidean space. Later chapters deal with trigonometry and differential geometry in Minkowski spaces. The book ends with a brief look at J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere. Minkowski Geometry will appeal to students and researchers interested in geometry, convexity theory and functional analysis.

The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that "in learning the sciences examples are of more use than precepts". We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a "global and analytical bias". We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincare duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hoelder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.

Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines. In this book, classical results from the cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra. It has also been used to give in sights into new developments in the spectral theory of linear operators. Various concepts from function theory and complex analytic geometry are drawn together and used to give a new approach to concrete spectral computations. The advantages of this approach are illustrated by a variety of examples, unexpected applications, and conceptually new ideas which should stimulate further research.