A theory of generalized Cauchy-Riemann systems with polar singularities of order not less than one is presented and its application to study of infinitesimal bending of surfaces having positive curvature and an isolated flat point is given. The book contains results of investigations obtained by the author and his collaborators.

Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions - like zigzag persistence and multiparameter persistence - and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.

There is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed, for instance, on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the Andre-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.

This book collects the papers of the conference held in Berlin, Germany, 27-29 August 2012, on 'Space, Geometry and the Imagination from Antiquity to the Modern Age'. The conference was a joint effort by the Max Planck Institute for the History of Science (Berlin) and the Centro die Ricerca Matematica Ennio De Giorgi (Pisa).

The behaviour of vanishing cycles is the cornerstone for understanding the geometry and topology of families of hypersurfaces, usually regarded as singular fibrations. This self-contained tract proposes a systematic geometro-topological approach to vanishing cycles, especially those appearing in non-proper fibrations, such as the fibration defined by a polynomial function. Topics which have been the object of active research over the past 15 years, such as holomorphic germs, polynomial functions, and Lefschetz pencils on quasi-projective spaces, are here shown in a new light: conceived as aspects of a single theory with vanishing cycles at its core. Throughout the book the author presents the current state of the art. Transparent proofs are provided so that non-specialists can use this book as an introduction, but all researchers and graduate students working in differential and algebraic topology, algebraic geometry, and singularity theory will find this book of great use.

The last half century has seen the development of many biological or physical t- ories that have explicitly or implicitly involved medial descriptions of objects and other spatial entities in our world. Simultaneously mathematicians have studied the properties of these skeletal descriptions of shape, and, stimulated by the many areas where medial models are useful, computer scientists and engineers have developed numerous algorithms for computing and using these models. We bring this kno- edge and experience together into this book in order to make medial technology more widely understood and used. The book consists of an introductory chapter, two chapters on the major mat- matical results on medial representations, ?ve chapters on algorithms for extracting medial models from boundary or binary image descriptions of objects, and three chapters on applications in image analysis and other areas of study and design. We hope that this book will serve the science and engineering communities using medial models and will provide learning material for students entering this ?eld. We are fortunate to have recruited many of the world leaders in medial theory, algorithms, and applications to write chapters in this book. We thank them for their signi?cant effort in preparing their contributions. We have edited these chapters and have combined them with the ?ve chapters that we have written to produce an integrated whole.

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.

Based on the conference/workshop on Continuum Theory and Dynamical Systems held in Lafayette, Louisiana, this reference illustrates the current expansion of knowledge on the relationship between these subjects. It presents new problems in hyperspaces, induced maps, universal maps, fixed-point sets, disconnected numbers and quotient maps.;Explaining the definitions and techniques used in the two fields and providing results from both areas, this volume: examines prime end (accessible) rotation numbers for chaotic sets and Henon maps; discussed the connection between the rotation shadowing property and the structure of the rotation set for annulus homeomorphisms; offers a Nielson-type theorum concerning the minimum number of fixed points for an area preserving homeomorphism of the two disc; constructs a closed unit disc that admits many inequivalent homeomorphisms that are Denjoy on the boundary and distinct irrational rotations on the interior; gives a geometric description of a horseshoe-type mapping of a plane disc into itself whose attracting set is not chainable; and considers semigroups generated by maps topologically conjugate to contractions.;Written by experts who provide a cross-disciplinary perspective, this volume is intended for applied mathematicians, topologists, geomesters, physicists and graduate-level students in these disciplines.

Written for researchers and developers applying Integrated Function Systems in the creation of fractal images, this book presents a modification of a widely used probabilistic algorithm for generating IFS-encoded images. The book also includes a discussion of how IFS techniques can be applied to produce animated motion pictures.

This book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.

There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.

Robert Langlands formulated his celebrated conjectures, initiating the Langlands Program, at the age of 31, profoundly changing the landscape of mathematics. Langlands, recipient of the Abel Prize, is famous for his insight in discovering links among seemingly dissimilar objects, leading to astounding results. This book is uniquely designed to serve a wide range of mathematicians and advanced students, showcasing Langlands' unique creativity and guiding readers through the areas of Langlands' work that are generally regarded as technical and difficult to penetrate. Part 1 features non-technical personal reflections, including Langlands' own words describing how and why he was led to formulate his conjectures. Part 2 includes survey articles of Langlands' early work that led to his conjectures, and centers on his principle of functoriality and foundational work on the Eisenstein series, and is accessible to mathematicians from other fields. Part 3 describes some of Langlands' contributions to mathematical physics.

Following on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals. Much of the material presented in this book has come to the fore in recent years. This includes methods for studying dimensions and other parameters of fractal sets and measures, as well as more sophisticated techniques such as the thermodynamic formalism and tangent measures. In addition to general theory, many examples and applications are described, in areas such as differential equations and harmonic analysis. The book is mathematically precise, but aims to give an intuitive feel for the subject, with underlying concepts described in a clear and accessible manner. The reader is assumed to be familiar with material from Fractal Geometry, but the main ideas and notation are reviewed in the first two chapters. Each chapter ends with brief notes on the development and current state of the subject. Exercises are included to reinforce the concepts. The author’s clear style and the up-to-date coverage of the subject make this book essential reading for all those who wish to develop their understanding of fractal geometry. Also available: Fractal Geometry: Mathematical Foundations and Applications Hardback ISBN 0-471-92287-0 Paperback ISBN 0-471-96777-7

Finite arrangements of convex bodies were intensively investigated in the second half of the twentieth century. Connections to many other subjects were made, including crystallography, the local theory of Banach spaces, and combinatorial optimisation. This book, the first one dedicated solely to the subject, provides an in-depth state-of-the-art discussion of the theory of finite packings and coverings by convex bodies. It contains various new results and arguments, besides collecting those scattered around in the literature, and provides a comprehensive treatment of problems whose interplay was not clearly understood before. In order to make the material more accessible, each chapter is essentially independent, and two-dimensional and higher-dimensional arrangements are discussed separately. Arrangements of congruent convex bodies in Euclidean space are discussed, and the density of finite packing and covering by balls in Euclidean, spherical and hyperbolic spaces is considered.

Based on a graduate course taught at Utrecht University, this book provides a short introduction to the theory of Foliations and Lie Groupoids to students who have already taken a first course in differential geometry. Ieke Moerdijk and Janez Mrcun include detailed references to enable students to find the requisite background material in the research literature. The text features many exercises and worked examples.

This is a revised and extended version of the popular first edition. Inspired by the work of Thom and Arnol'd on singularity theory, such topics as umbilics, ridges and subparabolic lines, all robust features of a smooth surface, which are rarely treated in elementary courses on differential geometry, are considered here in detail. These features are of immediate relevance in modern areas of application such as interpretation of range data from curved surfaces and the processing of magnetic resonance and cat-scan images. The text is based on extensive teaching at Liverpool University to audiences of advanced undergraduate and beginning postgraduate students in mathematics. However, the wide applicability of this material means that it will also appeal to scientists and engineers from a variety of other disciplines. The author has included many exercises and examples to illustrate the results proved.

One century after Hilbert constructed the first example of a non-classical affine plane, this book aims to summarize all the major results about geometries on surfaces. Acting both as a reference and a monograph, the authors have included detailed sections on what is known as well as outlining problems that remain to be solved. There are sections on classical geometries, methods for constructing non-classical geometries and classifications and characterizations of geometries. This work is related to a host of other fields including approximation, convexity, differential geometry topology and many more. This book will appeal to students, researchers and lecturers working in geometry or any one of the many associated areas outlined above.

The subject of this book is arithmetic algebraic geometry, an area between number theory and algebraic geometry. It is about applying geometric methods to the study of polynomial equations in rational numbers (Diophantine equations). This book represents the first complete and coherent exposition in a single volume, of both the theory and applications of torsors to rational points. Some very recent material is included. It is demonstrated that torsors provide a unified approach to several branches of the theory which were hitherto developing in parallel.

The original edition of The Geometry of Musical Rhythm was the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explained how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields. The book also introduced the distance approach to phylogenetic analysis and illustrated its application to the study of musical rhythm. The new edition retains all of this, while also adding 100 pages, 93 figures, 225 new references, and six new chapters covering topics such as meter and metric complexity, rhythmic grouping, expressive timbre and timing in rhythmic performance, and evolution phylogenetic analysis of ancient Greek paeonic rhythms. In addition, further context is provided to give the reader a fuller and richer insight into the historical connections between music and mathematics.

Peterson's egghead's Guide to Geometry will help students improve their understanding of the fundamental concepts of geometry. With the help of Peterson's new character, egghead, students can strengthen their math skills with narrative cartoons and graphics. Along the way there are plenty of study tips and exercises, making this the perfect guide for students struggling to improve their knowledge of geometry for standardized tests.
egghead's strategies and advice for improving geometry skillsFoundational geometry for students who need basic and remedial instructionDozens of sample exercises and solutions with loads of geometric figures and illustrationsEasy-to-read lessons with fun graphics that provide essential information and skills to help those students who learn visually

The Assouad dimension is a notion of dimension in fractal geometry that has been the subject of much interest in recent years. This book, written by a world expert on the topic, is the first thorough account of the Assouad dimension and its many variants and applications in fractal geometry and beyond. It places the theory of the Assouad dimension in context among up-to-date treatments of many key advances in fractal geometry, while also emphasising its diverse connections with areas of mathematics including number theory, dynamical systems, harmonic analysis, and probability theory. A final chapter detailing open problems and future directions for research brings readers to the cutting edge of this exciting field. This book will be an indispensable part of the modern fractal geometer's library and a valuable resource for pure mathematicians interested in the beauty and many applications of the Assouad dimension.