# curves and singularities a geometrical introduction to singularity theory

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## Curves And Singularities

Author : James William Bruce
ISBN : 0521429994
Genre : Mathematics
File Size : 66. 54 MB
Format : PDF, ePub, Docs

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.

## Introduction To Singularities And Deformations

Author : Gert-Martin Greuel
ISBN : 9783540284192
Genre : Mathematics
File Size : 47. 37 MB
Format : PDF, Docs

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.

## Singular Points Of Plane Curves

Author : C. T. C. Wall
ISBN : 0521547741
Genre : Mathematics
File Size : 63. 11 MB
Format : PDF, ePub

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.

## Singularities Of Smooth Functions And Maps

Author : J. Martinet
ISBN : 0521233984
Genre : Mathematics
File Size : 58. 16 MB
Format : PDF, ePub, Mobi

## The Theory Of Singularities And Its Applications

ISBN : 0521422809
Genre : Mathematics
File Size : 74. 5 MB
Format : PDF, ePub, Docs

This book describes those singularities encountered in different branches of mathematics. The distinguished mathematician, Vladimir Arnold, avoids giving difficult proofs of all the results in order to provide the reader with a concise and accessible overview of the many guises and areas in which singularities appear. Some of these areas include geometry and optics, optimal control theory and algebraic geometry, reflection groups theory, dynamical systems theory, and the classical and quantum catastrophe theory.

## Geometric Differentiation

Author : I. R. Porteous
ISBN : 0521002648
Genre : Mathematics
File Size : 25. 76 MB
Format : PDF, ePub, Mobi

This is a revised version of the popular Geometric Differentiation, first edition.

## Elementary Geometry Of Differentiable Curves

Author : C. G. Gibson
ISBN : 0521011078
Genre : Mathematics
File Size : 47. 50 MB
Format : PDF

An introductory text on the differential geometry of plane curves, first published in 2001.

## Algebraic Geometry And Statistical Learning Theory

Author : Sumio Watanabe
ISBN : 9780521864671
Genre : Computers
File Size : 49. 64 MB
Format : PDF, ePub

Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

## Resolution Of Singularities

Author : Steven Dale Cutkosky
ISBN : 9780821835555
Genre : Mathematics
File Size : 36. 74 MB
Format : PDF, Docs

The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.