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.

Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This collection of articles, authored by leading authorities in the field, comprises 16 expository essays presenting original research and expert surveys of important topics related to Riemann surfaces and their geometry. It complements the body of recorded research presented in the primary literature by broadening, re-working and extending it in a more focused and less formal framework, and provides a valuable commentary on contemporary work in the subject. An introductory section sets the scene and provides sufficient background to allow graduate students and research workers from other related areas access to the field.

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

Der vorliegende Klassiker bietet Studierenden und Forschenden in den Gebieten der Theoretischen und Mathematischen Physik eine ideale Einfuhrung in die Differentialgeometrie und Topologie. Beides sind wichtige Werkzeuge in den Gebieten der Astrophysik, der Teilchen- und Festkoerperphysik. Das Buch fuhrt durch: - Pfadintegralmethode und Eichtheorie - Mathematische Grundlagen von Abbildungen, Vektorraumen und Topologie - Fortgeschrittene Konzepte der Geometrie und Topologie und deren Anwendungen im Bereich der Flussigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie - Eine Zusammenfuhrung von Geometrie und Topologie: Faserbundel, charakteristische Klassen und Indextheoreme - Anwendungen von Geometrie und Topologie in der modernen Physik: Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer geometrischen Perspektive

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."

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.

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.

"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 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.

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.

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.

This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.

This book constitutes the refereed proceedings of the 12th IMA International Conference on the Mathematics of Surfaces, held in Sheffield, UK in September 2007. The papers cover a range of ideas from underlying theoretical tools to industrial uses of surfaces. Research is reported on theoretical aspects of surfaces as well as more practical topics.

This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to Teichmuller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation."

The papers collected in this volume are contributions to the 43rd session of the Seminaire de mathematiques superieures (SMS) on "Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology." This session took place at the Universite de Montreal in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most signi cant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are re ected in the present volume: Morse theory and related techniques in in nite dim- sional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian di?eomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all p- ticipants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane Belanger, our administrative assistant, for her help with the organi- tion and Mr. Andre Montpetit, our technical editor, for his help in the preparation of the volume."

This book contains an up-to-date survey and self-contained chapters on contact slant submanifolds and geometry, authored by internationally renowned researchers. The notion of slant submanifolds was introduced by Prof. B.Y. Chen in 1990, and A. Lotta extended this notion in the framework of contact geometry in 1996. Numerous differential geometers have since obtained interesting results on contact slant submanifolds. The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant -, hemi-slant -Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric f-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant submanifolds of almost paracontact manifolds. The book also includes interesting results on slant curves and magnetic curves, where the latter represents trajectories moving on a Riemannian manifold under the action of magnetic field. It presents detailed information on the most recent advances in the area, making it of much value to scientists, educators and graduate students.

This volume collects the papers accepted for presentation at the 11th IMA Conference on the Mathematics of Surfaces, held at Loughborough University, 5th-7th September 2005. As with all earlier conferences in the series, contri- tors to this volume comefrom manycountries. The paperspresented herere?ect the interest in a subject of relevance to mathematics, engineering, and computer science, especially in domains such as computer-aided design, computer vision, and computer graphics. The papers in the present volume include eight invited papers, as well as a larger number of submitted papers. They cover a range of ideas from - derlying theoretical tools to industrial and medical uses of surfaces. The latter category includes such diverse topics as surfaces in car design, and modelling of teeth, while the former includes papers on Voronoi diagrams, linear systems, estimation of curvatures on meshes, operators on meshes, intersection of sub- vision surfaces, approximate parameterization, condition numbers, Pythagorean hodographs, artifactsinB-splinesurfaces, B eziersurfacesofminimalenergy, line subdivision, subdivision surfaces, level sets and symmetry, the topology of - gebraic surfaces, curve analysis, interpolation with positivity, and conversion of cyclides to NURBS. Other papers concentrate on particular algorithms arising from applications, such as embedding graphs in manifolds, recoveryof 3D shape from shading, ?nding optimal feedrates for machining, detection of creases in range data, and ?lling holes in range data. We would like to thank all those who attended the conference and helped to make it a succes

This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.

In this book invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, certain time series, etc. Rather than dealing with the processes, the transition probabilities and the operators associated with these processes are studied.
Main features:
- an ergodic decomposition which is a "reference system" for dealing with ergodic measures
- "formulas" for the supports of invariant probability measures, some of which can be used to obtain algorithms for the graphical display of these supports
- helps to gain a better understanding of the structure of Markov-Feller operators, and, implicitly, of the discrete-time homogeneous Feller processes
- special efforts to attract newcomers to the theory of Markov processes in general, and to the topics covered in particular
- most of the results are new and deal with topics of intense research interest.

This book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.

The geometry of real submanifolds in complex manifolds and the analysis of their mappings belong to the most advanced streams of contemporary Mathematics. In this area converge the techniques of various and sophisticated mathematical fields such as P.D.E.s, boundary value problems, induced equations, analytic discs in symplectic spaces, complex dynamics. For the variety of themes and the surprisingly good interplaying of different research tools, these problems attracted the attention of some among the best mathematicians of these latest two decades. They also entered as a refined content of an advanced education. In this sense the five lectures of this volume provide an excellent cultural background while giving very deep insights of current research activity.

This book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

The volume develops the foundations of differential geometry so as to include finite-dimensional spaces with singularities and nilpotent functions, at the same level as is standard in the elementary theory of schemes and analytic spaces. The theory of differentiable spaces is developed to the point of providing a handy tool including arbitrary base changes (hence fibred products, intersections and fibres of morphisms), infinitesimal neighbourhoods, sheaves of relative differentials, quotients by actions of compact Lie groups and a theory of sheaves of Frechet modules paralleling the useful theory of quasi-coherent sheaves on schemes. These notes fit naturally in the theory of C DEGREES\infinity-rings and C DEGREES\infinity-schemes, as well as in the framework of Spallek's C DEGREES\infinity-standard differentiable spaces, and they require a certain familiarity with commutative algebra, sheaf theory, rings of differentiable functions and Frechet spaces."