OverthemillenniaDiophantineequationshavesuppliedanextremelyfertilesource ofproblems. Their study hasilluminated everincreasingpoints ofcontactbetween very di?erent subject areas, including algebraic geometry, mathematical logic, - godictheoryandanalyticnumber theory. Thefocus ofthis bookisonthe interface of algebraic geometry with analytic number theory, with the basic aim being to highlight the ro le that analytic number theory has to play in the study of D- phantine equations. Broadly speaking, analytic number theory can be characterised as a subject concerned with counting interesting objects. Thus, in the setting of Diophantine geometry, analytic number theory is especially suited to questions concerning the "distribution" of integral and rational points on algebraic varieties. Determining the arithmetic of a?ne varieties, both qualitatively and quantitatively, is much more complicated than for projective varieties. Given the breadth of the domain and the inherent di?culties involved, this book is therefore dedicated to an exp- ration of the projective setting. This book is based on a short graduate course given by the author at the I. C. T. P School and Conference on Analytic Number Theory, during the period 23rd April to 11th May, 2007. It is a pleasure to thank Professors Balasubra- nian, Deshouillers and Kowalski for organising this meeting. Thanks are also due to Michael Harvey and Daniel Loughran for spotting several typographical errors in an earlier draft of this book. Over the years, the author has greatly bene?ted fromdiscussing mathematicswithProfessorsde la Bret' eche,Colliot-Th' el' ene,F- vry, Hooley, Salberger, Swinnerton-Dyer and Wooley.

Leonardo da Pisa, perhaps better known as Fibonacci (ca. 1170 ca. 1240), selected the most useful parts of Greco-Arabic geometry for the book known as De Practica Geometrie. This translation offers a reconstruction of De Practica Geometrie as the author judges Fibonacci wrote it, thereby correcting inaccuracies found in numerous modern histories. It is a high quality translation with supplemental text to explain text that has been more freely translated. A bibliography of primary and secondary resources follows the translation, completed by an index of names and special words.

This book offers a comprehensive introduction to Subdivision Surface Modeling Technology focusing not only on fundamental theories but also on practical applications. It furthers readers' understanding of the contacts between spline surfaces and subdivision surfaces, enabling them to master the Subdivision Surface Modeling Technology for analyzing subdivision surfaces. Subdivision surface modeling is a popular technology in the field of computer aided design (CAD) and computer graphics (CG) thanks to its ability to model meshes of any topology. The book also discusses some typical Subdivision Surface Modeling Technologies, such as interpolation, fitting, fairing, intersection, as well as trimming and interactive editing. It is a valuable tool, enabling readers to grasp the main technologies of subdivision surface modeling and use them in software development, which in turn leads to a better understanding of CAD/CG software operations.

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.

Geometry is the cornerstone of computer graphics and computer animation, and provides the framework and tools for solving problems in two and three dimensions. This may be in the form of describing simple shapes such as a circle, ellipse, or parabola, or complex problems such as rotating 3D objects about an arbitrary axis.

Geometry for Computer Graphics draws together a wide variety of geometric information that will provide a sourcebook of facts, examples, and proofs for students, academics, researchers, and professional practitioners.

The book is divided into 4 sections: the first summarizes hundreds of formulae used to solve 2D and 3D geometric problems. The second section places these formulae in context in the form of worked examples. The third provides the origin and proofs of these formulae, and communicates mathematical strategies for solving geometric problems. The last section is a glossary of terms used in geometry.

This book features selected papers from The Seventh International Conference on Research and Education in Mathematics that was held in Kuala Lumpur, Malaysia from 25 - 27th August 2015. With chapters devoted to the most recent discoveries in mathematics and statistics and serve as a platform for knowledge and information exchange between experts from academic and industrial sectors, it covers a wide range of topics, including numerical analysis, fluid mechanics, operation research, optimization, statistics and game theory. It is a valuable resource for pure and applied mathematicians, statisticians, engineers and scientists, and provides an excellent overview of the latest research in mathematical sciences.

This volume is devoted to the "hyperbolic theory" of dynamical systems (DS), that is, the theory of smooth DS's with hyperbolic behaviour of the tra jectories (generally speaking, not the individual trajectories, but trajectories filling out more or less "significant" subsets in the phase space. Hyperbolicity the property that under a small displacement of any of a trajectory consists in point of it to one side of the trajectory, the change with time of the relative positions of the original and displaced points resulting from the action of the DS is reminiscent of the mot ion next to a saddle. If there are "sufficiently many" such trajectories and the phase space is compact, then although they "tend to diverge from one another" as it were, they "have nowhere to go" and their behaviour acquires a complicated intricate character. (In the physical literature one often talks about "chaos" in such situations. ) This type of be haviour would appear to be the opposite of the more customary and simple type of behaviour characterized by its own kind of stability and regularity of the motions (these words are for the moment not being used as a strict ter 1 minology but rather as descriptive informal terms). The ergodic properties of DS's with hyperbolic behaviour of trajectories (Bunimovich et al. 1985) have already been considered in Volume 2 of this series. In this volume we therefore consider mainly the properties of a topological character (see below 2 for further details)."

It is with pleasure that I write the foreword to this excellent book. A wide range of observations in geology and solid-earth geophysics can be - plained in terms of fractal distributions. In this volume a collection of - pers considers the fractal behavior of the Earth's continental crust. The book begins with an excellent introductory chapter by the editor Dr. V.P. Dimri. Surface gravity anomalies are known to exhibit power-law spectral behavior under a wide range of conditions and scales. This is self-affine fractal behavior. Explanations of this behavior remain controversial. In chapter 2 V.P. Dimri and R.P. Srivastava model this behavior using Voronoi tessellations. Another approach to understanding the structure of the continental crust is to use electromagnetic induction experiments. Again the results often exhibit power law spectral behavior. In chapter 3 K. Bahr uses a fractal based random resister network model to explain the observations. Other examples of power-law spectral observations come from a wide range of well logs using various logging tools. In chapter 4 M. Fedi, D. Fiore, and M. La Manna utilize multifractal models to explain the behavior of well logs from the main KTB borehole in Germany. In chapter 5 V.V. Surkov and H. Tanaka model the electrokinetic currents that may be as- ciated with seismic electric signals using a fractal porous media. In chapter 6 M. Pervukhina, Y. Kuwahara, and H. Ito use fractal n- works to correlate the elastic and electrical properties of porous media.

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.

Essentially, Orientations and Rotations treats the mathematical and computational foundations of texture analysis. It contains an extensive and thorough introduction to parameterizations and geometry of the rotation space. Since the notions of orientations and rotations are of primary importance for science and engineering, the book can be useful for a very broad audience using rotations in other fields.

This book is about modern algebraic geometry. The title "A Royal Road to Algebraic Geometry" is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work "Elements." Euclid is said to have answered: ""There is no royal road to geometry" "

The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense" there is" a royal road to algebraic geometry.

From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck's theory of schemes. Contemporary homological tools are explained.

The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime

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."

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.

This volume has grown from a conference entitled Harmonic Maps, Minimal Sur- faces and Geometric Flows which was held at the Universite de Bretagne Occi- dentale from July 7th-12th, 2002, in the town of Brest in Brittany, France. We welcomed many distinguished mathematicians from around the world and a dy- namic meeting took place, with many fruitful exchanges of ideas. In order to produce a work that would have lasting value to the mathematical community, the organisers decided to invite a small number of participants to write in-depth articles around a common theme. These articles provide a balance between introductory surveys and ones that present the newest results that lie at the frontiers of research. We thank these mathematicians, all experts in their field, for their contributions. Such meetings depend on the support of national organisations and the local community and we would like to thank the following: the Ministere de l'Education Nationale, Ministere des Affaires Etrangeres, Centre National de Recherche Sci en- tifique (CNRS), Conseil Regional de Bretagne, Conseil General du Finistere, Com- munaute Urbaine de Brest, Universite de Bretagne Occidentale (UBO), Faculte des Sciences de l'UBO, Laboratoire de Mathematiques de l'UBO and the Departement de Mathematiques de l'UBO. Their support was generous and ensured the success of the meeting. We would also like to thank the members of the scientific committee for their advice and for their participation in the conception and composition of this volume: Pierre Berard, Jean-Pierre Bourguignon, Frederic Helein, Seiki Nishikawa and Franz Pedit.

X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite des ecrivains au double alphabet, moi qui, il y a plus de quarante ans deja, avais accueilli sur mes terres un general epris de mathematiques. JI m'avait parle de ses projets grandioses en promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je suis entiche de geometrie et c'est d'elle dontje voudrais vous parler, oh! certes pas de toute la geometrie, mais de celle que fait l'artisan qui taille, burine, amene, gauchit, peaufine les formes. Mon interet pour le probleme dont je veux vous entretenir ici, je le dois a un ami ebeniste. En effet comme je rendais un jour visite il cet ami, je le trouvai dans son atelier affaire a un tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte de toupie et me dit: "Monsieur Besse, vous qui calculez les formes avec vos grimoires, que pensez-vous de ceci?)) Je le regardai interloque. Il poursuivit: "Regardez! Si vous prenez ce collier de laine et si vous le maintenez fermement avec un doigt place n'importe ou sur la toupie, eh bien! la toupie passera toujours juste en son interieur, sans laisser le moindre espace.)) Je rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une boule. Je me mis alors au travail ...

The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his coworkers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings proof of the Mordell Conjecture).This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required.

The problem of finding minimal surfaces, i. e. of finding the surface of least area among those bounded by a given curve, was one of the first considered after the foundation of the calculus of variations, and is one which received a satis factory solution only in recent years. Called the problem of Plateau, after the blind physicist who did beautiful experiments with soap films and bubbles, it has resisted the efforts of many mathematicians for more than a century. It was only in the thirties that a solution was given to the problem of Plateau in 3-dimensional Euclidean space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The methods of Douglas and Rado were developed and extended in 3-dimensions by several authors, but none of the results was shown to hold even for minimal hypersurfaces in higher dimension, let alone surfaces of higher dimension and codimension. It was not until thirty years later that the problem of Plateau was successfully attacked in its full generality, by several authors using measure-theoretic methods; in particular see De Giorgi [DG1, 2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren [AF1, 2]. Federer and Fleming defined a k-dimensional surface in IR" as a k-current, i. e. a continuous linear functional on k-forms. Their method is treated in full detail in the splendid book of Federer [FH 1].

Starting with the fundamentals of Q spaces and their relationships to Besov spaces, this book presents all major results around Q spaces obtained in the past 16 years. The applications of Q spaces in the study of the incompressible Navier-Stokes system and its stationary form are also discussed. This self-contained book can be used as an essential reference for researchers and graduates in analysis and partial differential equations.

This book contains 24 technical papers presented at the fourth edition of the Advances in Architectural Geometry conference, AAG 2014, held in London, England, September 2014. It offers engineers, mathematicians, designers, and contractors insight into the efficient design, analysis, and manufacture of complex shapes, which will help open up new horizons for architecture. The book examines geometric aspects involved in architectural design, ranging from initial conception to final fabrication. It focuses on four key topics: applied geometry, architecture, computational design, and also practice in the form of case studies. In addition, the book also features algorithms, proposed implementation, experimental results, and illustrations. Overall, the book presents both theoretical and practical work linked to new geometrical developments in architecture. It gathers the diverse components of the contemporary architectural tendencies that push the building envelope towards free form in order to respond to multiple current design challenges. With its introduction of novel computational algorithms and tools, this book will prove an ideal resource to both newcomers to the field as well as advanced practitioners.

Elementary number theory is concerned with arithmetic properties of the ring of integers. Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting, for example, analogues of the theorems of Fermat and Euler, Wilson¿s theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet¿s theorem on primes in an arithmetic progression. After presenting the required foundational material on function fields, the later chapters explore the analogy between global function fields and algebraic number fields. A variety of topics are presented, including: the ABC-conjecture, Artin¿s conjecture on primitive roots, the Brumer-Stark conjecture, Drinfeld modules, class number formulae, and average value theorems.