Curves and Singularities

A Geometrical Introduction to Singularity Theory


Author: James William Bruce,P. J. Giblin

Publisher: Cambridge University Press

ISBN: 9780521429993

Category: Mathematics

Page: 321

View: 8047

The differential geometry of curves and surfaces in Euclidean space has fascinated mathematicians since the time of Newton. Here the authors take a novel approach by casting the theory into a new light, that of singularity theory. The second edition of this successful textbook has been thoroughly revised throughout and includes a multitude of new exercises and examples. A new final chapter has been added that covers recently developed techniques in the classification of functions of several variables, a subject central to many applications of singularity theory. Also in this second edition are new sections on the Morse lemma and the classification of plane curve singularities. The only prerequisites for students to follow this textbook are a familiarity with linear algebra and advanced calculus. Thus it will be invaluable for anyone who would like an introduction to the modern theories of catastrophes and singularities.

Real and Complex Singularities


Author: Ana Claudia Nabarro,Juan J. Nuño-Ballesteros,Raúl Oset Sinha,Maria Aparecida Soares Ruas

Publisher: American Mathematical Soc.

ISBN: 1470422050

Category: Differential geometry -- Classical differential geometry -- Classical differential geometry

Page: 355

View: 6563

This volume is a collection of papers presented at the XIII International Workshop on Real and Complex Singularities, held from July 27–August 8, 2014, in São Carlos, Brazil, in honor of María del Carmen Romero Fuster's 60th birthday. The volume contains the notes from two mini-courses taught during the workshop: on intersection homology by J.-P. Brasselet, and on non-isolated hypersurface singularities and Lê cycles by D. Massey. The remaining contributions are research articles which cover topics from the foundations of singularity theory (including classification theory and invariants) to topology of singular spaces (links of singularities and semi-algebraic sets), as well as applications to topology (cobordism and Lefschetz fibrations), dynamical systems (Morse-Bott functions) and differential geometry (affine geometry, Gauss-maps, caustics, frontals and non-Euclidean geometries). This book is published in cooperation with Real Sociedad Matemática Española (RSME)

Topics on Real and Complex Singularities

Proceedings of the 4th Japanese–Australian Workshop (JARCS4)


Author: Satoshi Koike,Toshizumi Fukui,Laurentiu Paunescu,Adam Harris,Alexander Isaev

Publisher: World Scientific

ISBN: 9814596051

Category: Mathematics

Page: 212

View: 8867

A phenomenon which appears in nature, or human behavior, can sometimes be explained by saying that a certain potential function is maximized, or minimized. For example, the Hamiltonian mechanics, soapy films, size of an atom, business management, etc. In mathematics, a point where a given function attains an extreme value is called a critical point, or a singular point. The purpose of singularity theory is to explore the properties of singular points of functions and mappings. This is a volume on the proceedings of the fourth Japanese–Australian Workshop on Real and Complex Singularities held in Kobe, Japan. It consists of 11 original articles on singularities. Readers will be introduced to some important new notions for characterizations of singularities and several interesting results are delivered. In addition, current approaches to classical topics and state-of-the-art effective computational methods of invariants of singularities are also presented. This volume will be useful not only to the singularity theory specialists but also to general mathematicians. Contents:On the CR Hamiltonian Flows and CR Yamabe Problem (T Akahori)An Example of the Reduction of a Single Ordinary Differential Equation to a System, and the Restricted Fuchsian Relation (K Ando)Fronts of Weighted Cones (T Fukui and M Hasegawa)Involutive Deformations of the Regular Part of a Normal Surface (A Harris and K Miyajima)Connected Components of Regular Fibers of Differentiable Maps (J T Hiratuka and O Saeki)The Reconstruction and Recognition Problems for Homogeneous Hypersurface Singularities (A V Isaev)Openings of Differentiable Map-Germs and Unfoldings (G Ishikawa)Non Concentration of Curvature near Singular Points of Two Variable Analytic Functions (S Koike, T-C Kuo and L Paunescu)Saito Free Divisors in Four Dimensional Affine Space and Reflection Groups of Rank Four (J Sekiguchi)Holonomic Systems of Differential Equations of Rank Two with Singularities along Saito Free Divisors of Simple Type (J Sekiguchi)Parametric Local Cohomology Classes and Tjurina Stratifications for μ-Constant Deformations of Quasi-Homogeneous Singularities (S Tajima) Readership: Mathematicians in singularity theory or in adjacent areas; advanced undergraduates and graduate students in mathematics; non-experts interested in singularity theory and its applications. Key Features:Contains applications of the singularity theory to other mathematical fieldsNew topics in singularity theory, e.g. the relationship between free divisors and holonomic systems, openings of differentiable map-germs, non-concentration of curvatureIncludes articles by prize-winning researchers like Kimio Miyajima and Osamu SaekiKeywords:Singularities;CR Structure;Deformation Theory;Free Divisor;Concentration of Curvature;Holonomic System;Front;Opening

Singular Points of Plane Curves


Author: C. T. C. Wall

Publisher: Cambridge University Press

ISBN: 9780521547741

Category: Mathematics

Page: 370

View: 2457

Even the simplest singularities of planar curves, e.g. where the curve crosses itself, or where it forms a cusp, are best understood in terms of complex numbers. The full treatment uses techniques from algebra, algebraic geometry, complex analysis and topology and makes an attractive chapter of mathematics, which can be used as an introduction to any of these topics, or to singularity theory in higher dimensions. This book is designed as an introduction for graduate students and draws on the author's experience of teaching MSc courses; moreover, by synthesising different perspectives, he gives a novel view of the subject, and a number of new results.

Introduction to Singularities and Deformations


Author: Gert-Martin Greuel,Christoph Lossen,Eugenii I. Shustin

Publisher: Springer Science & Business Media

ISBN: 3540284192

Category: Mathematics

Page: 472

View: 3099

Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory and the theory of plane curve singularities. It includes complete proofs.

Singularities in Geometry and Topology

Proceedings of the Trieste Singularity Summer School and Workshop, ICTP, Trieste, Italy, 15 August - 3 September 2005


Author: Jean-Paul Brasselet

Publisher: World Scientific

ISBN: 981270681X

Category: Mathematical physics

Page: 916

View: 7112

Singularity theory appears in numerous branches of mathematics, as well as in many emerging areas such as robotics, control theory, imaging, and various evolving areas in physics. The purpose of this proceedings volume is to cover recent developments in singularity theory and to introduce young researchers from developing countries to singularities in geometry and topology. The contributions discuss singularities in both complex and real geometry. As such, they provide a natural continuation of the previous school on singularities held at ICTP (1991), which is recognized as having had a major influence in the field.

Ebene algebraische Kurven


Author: Egbert Brieskorn,Horst Knörrer

Publisher: N.A


Category: Mathematics

Page: 964

View: 1755

Nonlinear Computational Geometry


Author: Ioannis Z. Emiris,Frank Sottile,Thorsten Theobald

Publisher: Springer Science & Business Media

ISBN: 1441909990

Category: Mathematics

Page: 239

View: 4752

An original motivation for algebraic geometry was to understand curves and surfaces in three dimensions. Recent theoretical and technological advances in areas such as robotics, computer vision, computer-aided geometric design and molecular biology, together with the increased availability of computational resources, have brought these original questions once more into the forefront of research. One particular challenge is to combine applicable methods from algebraic geometry with proven techniques from piecewise-linear computational geometry (such as Voronoi diagrams and hyperplane arrangements) to develop tools for treating curved objects. These research efforts may be summarized under the term nonlinear computational geometry. This volume grew out of an IMA workshop on Nonlinear Computational Geometry in May/June 2007 (organized by I.Z. Emiris, R. Goldman, F. Sottile, T. Theobald) which gathered leading experts in this emerging field. The research and expository articles in the volume are intended to provide an overview of nonlinear computational geometry. Since the topic involves computational geometry, algebraic geometry, and geometric modeling, the volume has contributions from all of these areas. By addressing a broad range of issues from purely theoretical and algorithmic problems, to implementation and practical applications this volume conveys the spirit of the IMA workshop.



Author: Unité de recherche associée au CNRS' Geometry, Analysis, and Topology

Publisher: Cambridge University Press

ISBN: 9780521466318

Category: Mathematics

Page: 419

View: 6183

This book contains papers given at the International Singularity Conference held in 1991 at Lille.

Visual Motion of Curves and Surfaces


Author: Roberto Cipolla,Peter Giblin

Publisher: Cambridge University Press

ISBN: 9780521632515

Category: Computers

Page: 184

View: 3558

Computer vision aims to detect and reconstruct features of surfaces from the images produced by cameras, in some way mimicking the way in which humans reconstruct features of the world around them by using their eyes. In this book the authors describe research in computer vision aimed at recovering the 3D shape of surfaces from image sequences of their 'outlines'. They provide all the necessary background in differential geometry (assuming knowledge of elementary algebra and calculus) and in the analysis of visual motion, emphasising intuitive visual understanding of the geometric techniques with computer-generated illustrations. They also give a thorough introduction to the mathematical techniques and the details of the implementations and apply the methods to data from real images using the most current techniques.

Singularity Theory


Author: Denis Cheniot,Jean-Paul Brasselet

Publisher: World Scientific

ISBN: 9812707492

Category: Geometry, Differential

Page: 1065

View: 959

The Singularity School and Conference took place in Luminy, Marseille, from January 24th to February 25th 2005. More than 180 mathematicians from over 30 countries converged to discuss recent developments in singularity theory. The volume contains the elementary and advanced courses conducted by singularities specialists during the conference, general lectures on singularity theory, and lectures on applications of the theory to various domains. The subjects range from geometry and topology of singularities, through real and complex singularities, to applications of singularities.

Elementare und algebraische Zahlentheorie

Ein moderner Zugang zu klassischen Themen


Author: Stefan Müller-Stach,Jens Piontkowski

Publisher: Springer-Verlag

ISBN: 3834882631

Category: Mathematics

Page: 261

View: 3815

Das Buch wendet sich an alle, die in die klassischen Themen der Zahlentheorie einsteigen wollen. Viel Wert wird auf die konkrete Berechenbarkeit bei allen Problemlösungen gelegt. So gibt es auch Abschnitte über moderne Primzahltests und Faktorisierungsalgorithmen und am Ende des Buches wird ein Weg zur Bestimmung der Klassenzahl der quadratischen Zahlkörper aufgezeigt. Im Rahmen der Bachelor-/Master-Studiengänge eignet sich das Buch als Grundlage für zwei Semester: ein Aufbaumodul in elementarer Zahlentheorie mit einem Vertiefungsmodul in algebraischer Zahlentheorie.