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.

Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introduction to rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kahler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction to the topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists; namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bäcklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bäcklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.

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 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.

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 graduate level text covers an exciting and active area of research at the crossroads of several different fields in Mathematics and Physics. In Mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in Physics String Theory and Mirror Symmetry. Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kahler structures suitable for beginning graduate students, the text covers seminal results such as Yau's proof of the Calabi Conjecture, and takes the reader all the way to the frontiers of current research in calibrated geometry, giving many open problems.

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

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 contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold. This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface. The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory. This work is intended for advanced students in mathematical physics and researchers alike.

Since the time of Lagrange and Euler, it has been well known that an understanding of algebraic curves can illuminate the picture of rigid bodies provided by classical mechanics. Many mathematicians have established a modern view of the role played by algebraic geometry in recent years. This book presents some of these modern techniques, which fall within the orbit of finite dimensional integrable systems. The main body of the text presents a rich assortment of methods and ideas from algebraic geometry prompted by classical mechanics, while in appendices the author describes general, abstract theory. She gives the methods a topological application, for the first time in book form, to the study of Liouville tori and their bifurcations.

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."

This is an introduction to noncommutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fiber bundles is assumed, making this book accessible to graduate students and newcomers to this field.

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.

Singularity theory is a broad subject with vague boundaries. It draws on many other areas of mathematics, and in turn has contributed to many areas both within and outside mathematics, in particular differential and algebraic geometry, knot theory, differential equations, bifurcation theory, Hamiltonian mechanics, optics, robotics and computer vision. This volume consists of two dozen articles from some of the best known figures in singularity theory, and it presents an up-to-date survey of research in this area.

Here is an introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences. The book is well illustrated and contains several hundred worked examples and exercises. From the familiar lines and conics of elementary geometry the reader proceeds to general curves in the real affine plane, with excursions to more general fields to illustrate applications, such as number theory. By adding points at infinity the affine plane is extended to the projective plane, yielding a natural setting for curves and providing a flood of illumination into the underlying geometry. A minimal amount of algebra leads to the famous theorem of Bezout, while the ideas of linear systems are used to discuss the classical group structure on the cubic.

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."

This 1998 book is both an introduction to, and a survey of, some topics of singularity theory; in particular the studying of singularities by means of differential forms. Here some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and the theory of period maps, are developed in the local situation to study the case of isolated singularities of holomorphic functions. The author introduces the Gauss-Manin connection on the vanishing cohomology of a singularity, that is on the cohomology fibration associated to the Milnor fibration, and draws on the work of Brieskorn and Steenbrink to calculate this connection, and the limit mixed Hodge structure. This will be an excellent resource for all researchers whose interests lie in singularity theory, and algebraic or differential geometry.

This volume collects the papers accepted for presentation at the 10th IMA C- ference on the Mathematics of Surfaces, held at the University of Leeds, UK, September 15-17, 2003. As with all earlier conferences in the series, contributors to this volume come from a wide variety of countries in Asia, Europe and North America. The papers presented here re?ect the continued relevance of the s- ject, and, by comparison with the contents of earlier volumes in the series, the changing nature of application areas and mathematical techniques. To give the casual browser of these proceedings an idea of the diversity of the subject area, the papers in the present volume cover such topics and techniques as digital geometry processing, computer graphics, surface topology, medical applications, subdivision surfaces, surface reconstruction, surface triangulation, waterma- ing, data compression, data smoothing, human-computer interaction, extracting shapeformshading,surfaceheightrecovery,reverseengineering,box-splines,the Plateau Problem, splines (a variety of papers), trans?nite blending, and a?ne arithmetic. There is also a paper in memory of the late Prof. Josef Hoschek of theTechnischeUniversit. atDarmstadt,co-authoredbyaformerresearchstudent, Prof. Bert Juttler, .. on the subject of using line congruences for parameterizing special algebraic surfaces. We would like to thank all those who attended the conference and helped to makeitasuccess. Inparticularwethankthosewhocontributedtotheseproce- ings.

A concise introduction to the techniques used to prove the Baum-Connes conjecture. The Baum-Connes conjecture predicts that the K-homology of the reduced C DEGREES*-algebra of a group can be computed as the equivariant K-homology of the classifying space for proper actions. The approach is expository, but it contains proofs of many basic results on topological K-homology and the K-theory of C DEGREES*-algebras. It features a detailed introduction to Bredon homology for infinite groups, with applications to K-homology. It also contains a detailed discussion of naturality questions concerning the assembly map, a topic not well documented in the literature.

The book is aimed at advanced graduate students and researchers in the area, leading to current research problems.

Tight and taut manifolds form an important and special class of surfaces within differential geometry. This book contains in-depth articles by experts in the field as well as an extensive and comprehensive bibliography. This survey will open new avenues for further research and will be an important addition to any geometer's library.

* Devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow.

* Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics.

These notes deal with deformation theory of complex analytic singularities and related objects. The first part treats general theory. The central notion is that of versal deformationin several variants. The theory is developed both in an abstract way and in a concrete way suitable for computations. The second part deals with more specific problems, specially on curves and surfaces. Smoothings of singularities are the main concern. Examples are spread throughout the text.