To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Bruning and V.W. Guillemin. The point of departure are two relatively short but very remarkable papers be Henry Cartan, published in 1950 in the Proceedings of the "Colloque de Topologie." These papers are reproduced here, together with a modern introduction to the subject, written by two of the leading experts in the field. This "introduction" comes as a textbook of its own, though, presenting the first full treatment of equivariant cohomology in the de Rahm setting. The well known topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects, leading up to the localization theorems and other very recent results."

This book by two of the foremost researchers and writers in the field is the first part of a treatise that covers the subject in breadth and depth, paying special attention to the historical origins of the theory. Both individually and collectively these volumes have already become standard references.

In this book the authors develop and work out applications to gravity and gauge theories and their interactions with generic matter fields, including spinors in full detail. Spinor fields in particular appear to be the prototypes of truly gauge-natural objects, which are not purely gauge nor purely natural, so that they are a paradigmatic example of the intriguing relations between gauge natural geometry and physical phenomenology. In particular, the gauge natural framework for spinors is developed in this book in full detail, and it is shown to be fundamentally related to the interaction between fermions and dynamical tetrad gravity.

The book covers topics in the theory of algebraic transformation groups and algebraic varieties which are very much at the frontier of mathematical research.

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum."

State-of-the-art in qualitative theory of functional differential equations; Most of the new material has never appeared in book form and some not even in papers; Second edition updated with new topics and results; Methods discussed will apply to other equations and applications

This monograph contains an exposition of the theory of minimal surfaces in Euclidean space, with an emphasis on complete minimal surfaces of finite total curvature. Our exposition is based upon the philosophy that the study of finite total curvature complete minimal surfaces in R3, in large measure, coincides with the study of meromorphic functions and linear series on compact Riemann sur faces. This philosophy is first indicated in the fundamental theorem of Chern and Osserman: A complete minimal surface M immersed in R3 is of finite total curvature if and only if M with its induced conformal structure is conformally equivalent to a compact Riemann surface Mg punctured at a finite set E of points and the tangential Gauss map extends to a holomorphic map Mg _ P2. Thus a finite total curvature complete minimal surface in R3 gives rise to a plane algebraic curve. Let Mg denote a fixed but otherwise arbitrary compact Riemann surface of genus g. A positive integer r is called a puncture number for Mg if Mg can be conformally immersed into R3 as a complete finite total curvature minimal surface with exactly r punctures; the set of all puncture numbers for Mg is denoted by P (M ). For example, Jorge and Meeks JM] showed, by constructing an example g for each r, that every positive integer r is a puncture number for the Riemann surface pl."

General relativity ranks among the most accurately tested fundamental theories in all of physics. Deficiencies in mathematical and conceptual understanding still exist, hampering further progress. This book collects surveys by experts in mathematical relativity writing about the current status of, and problems in, their fields. There are four contributions for each of the following mathematical areas: differential geometry and differential topology, analytical methods and differential equations, and numerical methods.

Here is a systematic approach to such fundamental questions as: What mathematical structures does Einstein-Weyl causality impose on a point-set that has no other previous structure defined on it? The author proposes an axiomatization of the physics inspired notion of Einstein-Weyl causality and investigating the consequences in terms of possible topological spaces. One significant result is that the notion of causality can effectively be extended to discontinuum.

This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered (the Koszul formula in disguise); an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.

Supermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of differential forms. For super Lie groups the standard results are shown, including the construction of a super Lie group for any super Lie algebra. The last chapter is entirely devoted to super connections.

The book requires standard undergraduate knowledge on super differential geometry and super Lie groups.

This overview is based on the talk [101] given at the mini-workshop 0648c "Dirac operators in di?erential and non-commutative geometry", Mat- matisches Forschungsinstitut Oberwolfach. Intended for non-specialists, it explores the spectrum of the fundamental Dirac operator on Riemannian spin manifolds, including recent research and open problems. No background in spin geometry is required; nevertheless the reader is assumed to be fam- iar with basic notions of di?erential geometry (manifolds, Lie groups, vector and principal bundles, coverings, connections, and di?erential forms). The surveys [41, 132], which themselves provide a very good insight into closed manifolds, served as the starting point. We hope the content of this book re?ects the wide range of ?ndings on and sometimes amazing applications of the spin side of spectral theory and will attract a new audience to the topic. vii Acknowledgements TheauthorwouldliketothanktheMathematischesForschungsinstitutOb- wolfach for its friendly hospitality and stimulating atmosphere, as well as the organizers and all those who participated in the mini-workshop. This s- vey would not have been possible without the encouragement and advice of Christian B.. ar and Oussama Hijazi as well as the support of the German Research Foundation's Sonderforschungsbereich 647 "Raum, Zeit, Materie. Analytische und geometrische Strukturen" (Collaborative Research Center 647/Space, Time and Matter. Analytical and Geometric Structures). We would also like to thank Bernd Ammann and Nadine Grosse for their enlig- eningdiscussions and usefulreferences.

Finsler geometry generalizes Riemannian geometry in the same sense that Banach spaces generalize Hilbert spaces. This book presents an expository account of seven important topics in Riemann-Finsler geometry, ones which have recently undergone significant development but have not had a detailed pedagogical treatment elsewhere. Each article will open the door to an active area of research, and is suitable for a special topics course in graduate-level differential geometry. The contributors consider issues related to volume, geodesics, curvature, complex differential geometry, and parametrized jet bundles, and include a variety of instructive examples.

Linear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat eq- tion or,equivalently,the membraneequation. Forthis intensivelywell-studiedlinear problem there are two main lines of results. The ?rst line consists of existence and regularity results. Usually the solution exists and "gains two orders of differen- ation" with respect to the source term. The second line contains comparison type results, namely the property that a positive source term implies that the solution is positive under suitable side constraints such as homogeneous Dirichlet bou- ary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of results hold for general second order elliptic problems, see the books by Gilbarg-Trudinger [198] and Protter-Weinberger [347]. For linear higher order elliptic problems the existence and regularitytype results - main, as one may say, in their full generality whereas comparison type results may fail. Here and in the sequel "higher order" means order at least four. Most interesting models, however, are nonlinear. By now, the theory of second order elliptic problems is quite well developed for semilinear, quasilinear and even for some fully nonlinear problems. If one looks closely at the tools being used in the proofs, then one ?nds that many results bene?t in some way from the positivity preserving property. Techniques based on Harnack's inequality, De Giorgi-Nash- Moser's iteration, viscosity solutions etc.

S.G. Gindikin, I.I. Pjateckii-Sapiro, E.B. Vinberg: Homogeneous K hler manifolds.- S.G. Greenfield: Extendibility properties of real submanifolds of Cn.- W. Kaup: Holomorphische Abbildungen in Hyperbolische R ume.- A. Koranyi: Holomorphic and harmonic functions on bounded symmetric domains.- J.L. Koszul: Formes harmoniques vectorielles sur les espaces localement sym triques.- S. Murakami: Plongements holomorphes de domaines sym triques.- E.M. Stein: The analogues of Fatous 's theorem and estimates for maximal functions.

This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

Abstract topological tools from generalized metric spaces are applied in this volume to the construction of locally uniformly rotund norms on Banach spaces. The book offers new techniques for renorming problems, all of them based on a network analysis for the topologies involved inside the problem.

Maps from a normed space X to a metric space Y, which provide locally uniformly rotund renormings on X, are studied and a new frame for the theory is obtained, with interplay between functional analysis, optimization and topology using subdifferentials of Lipschitz functions and covering methods of metrization theory. Any one-to-one operator T from a reflexive space X into c0 (T) satisfies the authors' conditions, transferring the norm to X. Nevertheless the authors' maps can be far from linear, for instance the duality map from X to X* gives a non-linear example when the norm in X is Frechet differentiable.

This volume will be interesting for the broad spectrum of specialists working in Banach space theory, and for researchers in infinite dimensional functional analysis."

In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the Lp spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L2-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.

"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory." Mathematical Reviews

This volume contains papers by the main participants in the meeting of the 6th International Colloquium on Differential Geometry and its Related Fields (ICDG2018).The volume consists of papers devoted to the study of recent topics in geometric structures on manifolds - which are related to complex analysis, symmetric spaces and surface theory - and also in discrete mathematics.Thus, it presents a broad overview of differential geometry and provides up-to-date information to researchers and young scientists in this field, and also to those working in the wide spectrum of mathematics.

This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, K hler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac 's quantization of the electrical charge.

This volume contains the notes of lectures given at the School "Quantum Potential Theory: Structure and Applications to Physics." This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald from February 26 to March 9, 2007. We thank the lecturers for the hard work they - complished in preparing and giving these lectures and in writing these notes. Theirlecturesgiveanintroductiontocurrentresearchintheirdomains, which is essentially self-contained and should be accessible to Ph.D. students. We hope that this volume will help to bring together researchers from the areas of classical and quantum probability, functional analysis and operator al- bras, and theoretical and mathematical physics, and contribute in this way to developing further the subject of quantum potential theory. We are greatly indebted to the Alfried Krupp von Bohlen und Halbach- Stiftung for the ?nancial support, without which the school would not have been possible. We are also very thankful for the support by the University of Greifswald and the University of Franche-Comt e.One of the organisers(UF) wassupportedby a MarieCurieOutgoingInternationalFellowshipofthe EU (Contract Q-MALL MOIF-CT-2006-022137). Special thanks go to Melanie Hinz who helped with the preparation and organisation of the school and who took care of all of the logistics. Finally, we would like to thank all the students for coming to Greifswald and helping to make the school a success."

The main motivation for this book lies in the breadth of applications in which a statistical model is used to represent small departures from, for example, a Poisson process. Our approach uses information geometry to provide a c- mon context but we need only rather elementary material from di?erential geometry, information theory and mathematical statistics. Introductory s- tions serve together to help those interested from the applications side in making use of our methods and results. We have available Mathematica no- books to perform many of the computations for those who wish to pursue their own calculations or developments. Some 44 years ago, the second author ?rst encountered, at about the same time, di?erential geometry via relativity from Weyl's book [209] during - dergraduate studies and information theory from Tribus [200, 201] via spatial statistical processes while working on research projects at Wiggins Teape - searchandDevelopmentLtd-cf. theForewordin[196]and[170,47,58]. H- ing started work there as a student laboratory assistant in 1959, this research environment engendered a recognition of the importance of international c- laboration, and a lifelong research interest in randomness and near-Poisson statistical geometric processes, persisting at various rates through a career mainly involved with global di?erential geometry. From correspondence in the 1960s with Gabriel Kron [4, 124, 125] on his Diakoptics, and with Kazuo Kondo who in?uenced the post-war Japanese schools of di?erential geometry and supervised Shun-ichi Amari's doctorate [6], it was clear that both had a much wider remit than traditionally pursued elsewhere.

Radiolaria are a very diverse marine siliceous microplankton group that have existed at least snice the Cambrian to the recent. This volume gives a representative view of research topics discussed at the 10th International Meeting of Radiolarian Palaeontologists. The articles of this volume cover mainly radiolarian biochronology and radiolarian fauna changes.