Descriptive topology and functional analysis, with extensive material demonstrating new connections between them, are the subject of the first section of this work. Applications to spaces of continuous functions, topological Abelian groups, linear topological equivalence and to the separable quotient problem are included and are presented as open problems. The second section is devoted to Banach spaces, Banach algebras and operator theory. Each chapter presents a lot of worthwhile and important recent theorems with an abstract discussing the material in the chapter. Each chapter can almost be seen as a survey covering a particular area.

The propagation of acoustic and electromagnetic waves in stratified media is a subject that has profound implications in many areas of applied physics and in engineering, just to mention a few, in ocean acoustics, integrated optics, and wave guides. See for example Tolstoy and Clay 1966, Marcuse 1974, and Brekhovskikh 1980. As is well known, stratified media, that is to say media whose physical properties depend on a single coordinate, can produce guided waves that propagate in directions orthogonal to that of stratification, in addition to the free waves that propagate as in homogeneous media. When the stratified media are perturbed, that is to say when locally the physical properties of the media depend upon all of the coordinates, the free and guided waves are no longer solutions to the appropriate wave equations, and this leads to a rich pattern of wave propagation that involves the scattering of the free and guided waves among each other, and with the perturbation. These phenomena have many implications in applied physics and engineering, such as in the transmission and reflexion of guided waves by the perturbation, interference between guided waves, and energy losses in open wave guides due to radiation. The subject matter of this monograph is the study of these phenomena.

Derived from the author's course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology, algebraic/differential geometry, and Lie groups. The first two chapters review differential and integral calculus of several variables and present fundamental results that are used throughout the text. The next few chapters focus on smooth manifolds as submanifolds in a Euclidean space, the algebraic machinery of differential forms necessary for studying integration on manifolds, abstract smooth manifolds, and the foundation for homotopical aspects of manifolds. The author then discusses a central theme of the book: intersection theory. He also covers Morse functions and the basics of Lie groups, which provide a rich source of examples of manifolds. Exercises are included in each chapter, with solutions and hints at the back of the book. A sound introduction to the theory of smooth manifolds, this text ensures a smooth transition from calculus-level mathematical maturity to the level required to understand abstract manifolds and topology. It contains all standard results, such as Whitney embedding theorems and the Borsuk-Ulam theorem, as well as several equivalent definitions of the Euler characteristic.

This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22-26, 2016 and Probabilistic Methods in Topology, held from November 14-18, 2016 at the Centre de Recherches Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrodinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.

In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics.

If you have not heard about cohomology, The Heart of Cohomology may be suited for you. The book gives Fundamental notions in cohomology for examples, functors, representable functors, Yoneda embedding, derived functors, spectral sequences, derived categories are explained in elementary fashion. Applications to sheaf cohomology. In addition, the book examines cohomological aspects of D-modules and of the computation of zeta functions of the Weierstrass family.

Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e.

The book gives a comprehensive account of the basic algebraic properties of the classical groups over rings. Much of the theory appears in book form for the first time, and most proofs are given in detail. The book also includes a revised and expanded version of DieudonnA(c)'s classical theory over division rings. The authors analyse congruence subgroups, normal subgroups and quotient groups, they describe their isomorphisms and investigate connections with linear and hermitian K-theory. A first insight is offered through the simplest case of the general linear group. All the other classical groups, notably the symplectic, unitary and orthogonal groups, are dealt with uniformly as isometry groups of generalized quadratic modules. New results on the unitary Steinberg groups, the associated K2-groups and the unitary symbols in these groups lead to simplified presentation theorems for the classical groups. Related material such as the K-theory exact sequences of Bass and Sharpe and the Merkurjev-Suslin theorem is outlined. "From" "the foreword by J. DieudonnA(c): " "All mathematicians interested in classical groups should be grateful to these two outstanding investigators for having brought together old and new results (many of them their own) into a superbly organized whole. I am confident that their book will remain for a long time the standard reference in the theory."

This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry.

The aim of this book is to present recently discovered connections between Artin's braid groups En and left self-distributive systems (also called LD systems), which are sets equipped with a binary operation satisfying the left self-distributivity identity x(yz) = (xy)(xz). (LD) Such connections appeared in set theory in the 1980s and led to the discovery in 1991 of a left invariant linear order on the braid groups. Braids and self-distributivity have been studied for a long time. Braid groups were introduced in the 1930s by E. Artin, and they have played an increas ing role in mathematics in view of their connection with many fields, such as knot theory, algebraic combinatorics, quantum groups and the Yang-Baxter equation, etc. LD-systems have also been considered for several decades: early examples are mentioned in the beginning of the 20th century, and the first general results can be traced back to Belousov in the 1960s. The existence of a connection between braids and left self-distributivity has been observed and used in low dimensional topology for more than twenty years, in particular in work by Joyce, Brieskorn, Kauffman and their students. Brieskorn mentions that the connection is already implicit in (Hurwitz 1891). The results we shall concentrate on here rely on a new approach developed in the late 1980s and originating from set theory."

This book is a collection of articles from several world-class researchers, and is inspired by Sir Roger Penrose's work. It gives an overview of the interaction between geometry and physics, from which many important developments have emerged. The volume collects together ideas from across the physical sciences, and indicates the many applications of geometrical ideas and techniques across mathematics and mathematical physics.

Since the year 2000, we have witnessed several outstanding results in geometry that have solved long-standing problems such as the Poincare conjecture, the Yau-Tian-Donaldson conjecture, and the Willmore conjecture. There are still many important and challenging unsolved problems including, among others, the Strominger-Yau-Zaslow conjecture on mirror symmetry, the relative Yau-Tian-Donaldson conjecture in Kahler geometry, the Hopf conjecture, and the Yau conjecture on the first eigenvalue of an embedded minimal hypersurface of the sphere. For the younger generation to approach such problems and obtain the required techniques, it is of the utmost importance to provide them with up-to-date information from leading specialists.The geometry conference for the friendship of China and Japan has achieved this purpose during the past 10 years. Their talks deal with problems at the highest level, often accompanied with solutions and ideas, which extend across various fields in Riemannian geometry, symplectic and contact geometry, and complex geometry.

The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers)."

The two-volume work is intended to function as a textbook for graduate students in economics as well as a reference work for economic scholars. Assuming only the minimal mathematics background required of every second-year graduate student in economics, these two volumes provide a self-contained and careful development of mathematics through locally convex topological vector spaces, and fixed-point, separation, and selection theorems in such spaces. Volume One covers basic set theory, sequences and series, continuous and semi-continuous functions, an introduction to general linear spaces, basic convexity theory, and applications to economics. Volume Two introduces general topology, the theory of correspondences on and into topological spaces, Banach spaces, topological vector spaces, and maximum, fixed-point, and selection theorems for such spaces.

This volume contains 17 surveys that cover many recent developments in Discrete Geometry and related fields. Besides presenting the state-of-the-art of classical research subjects like packing and covering, it also offers an introduction to new topological, algebraic and computational methods in this very active research field. The readers will find a variety of modern topics and many fascinating open problems that may serve as starting points for research.

This two-volume set containts parts I and II. Each volume is a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions are split into two parts: ``Integrable Systems'' and ``Quantum Theories and Algebraic Geometry'', reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

This is a comprehensive introduction into the method of inverse spectra - a powerful method successfully employed in various branches of topology.

The notion of an inverse sequence and its limits, first appeared in the well-known memoir by Alexandrov where a special case of inverse spectra - the so-called projective spectra - were considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz. Meanwhile, Freudental, had introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works.

Subsequently, inverse spectra began to be widely studied and applied, not only in the various major branches of topology, but also in functional analysis and algebra. This is not surprising considering the categorical nature of inverse spectra and the extraordinary power of the related techniques.

Updated surveys (including proofs of several statements) of the Hilbert cube and Hilbert space manifold theories are included in the book. Recent developments of the Menger and Nobeling manifold theories are also presented.

This work significantly extends and updates the author's previously published book and has been completely rewritten in order to incorporate new developments in the field.

The topics in this survey volume concern research done on the differential geom etry of foliations over the last few years. After a discussion of the basic concepts in the theory of foliations in the first four chapters, the subject is narrowed down to Riemannian foliations on closed manifolds beginning with Chapter 5. Following the discussion of the special case of flows in Chapter 6, Chapters 7 and 8 are de voted to Hodge theory for the transversal Laplacian and applications of the heat equation method to Riemannian foliations. Chapter 9 on Lie foliations is a prepa ration for the statement of Molino's Structure Theorem for Riemannian foliations in Chapter 10. Some aspects of the spectral theory for Riemannian foliations are discussed in Chapter 11. Connes' point of view of foliations as examples of non commutative spaces is briefly described in Chapter 12. Chapter 13 applies ideas of Riemannian foliation theory to an infinite-dimensional context. Aside from the list of references on Riemannian foliations (items on this list are referred to in the text by [ ]), we have included several appendices as follows. Appendix A is a list of books and surveys on particular aspects of foliations. Appendix B is a list of proceedings of conferences and symposia devoted partially or entirely to foliations. Appendix C is a bibliography on foliations, which attempts to be a reasonably complete list of papers and preprints on the subject of foliations up to 1995, and contains approximately 2500 titles.

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES."

Turbulence theory is one of the most intriguing parts of fluid mechanics and many outstanding scientists have tried to apply their knowledge to the development of the theory and to offer useful recommendations for solution of some practical problems. In this monograph the author attempts to integrate many specific approaches into the unified theory. The basic premise is the simple idea that a small eddy, that is an element of turbulent meso-structure, possesses its own dynamics as an object rotating with its own spin velocity and obeying the Newton dynamics of a finite body. A number of such eddies fills a coordinate cell, and the angular momentum balance has to be formulated for this spatial cell. If the cell coincides with a finite difference element at a numerical calculation and if the external length scale is large, this elementary volume can be considered as a differential one and a continuum parameterization has to be used. Nontrivial angular balance is a consequence of the asymmetrical Reynolds stress action at the oriented sides of an elementary volume. At first glance, the averaged dyad of velocity components is symmetrical, == However, if averaging is performed over the plane with normal nj, the principle of commutation is lost. As a result, the stress tensor asymmetry j is determined by other factors that participate in the angular momentum balance. This is the only possibility to determine a stress in engineering."

Karl Menger, one of the founders of dimension theory, belongs to the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna, and after his emigration to the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced some of his colleagues. The Selecta Mathematica contain Menger's major mathematical papers, based on his own selection of his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, as well as writings on economics, sociology, logic, philosophy and mathematical results. The two volumes are a monument to the diversity and originality of Menger's ideas.