Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T06:38:15.496Z Has data issue: false hasContentIssue false

Introduction

Published online by Cambridge University Press:  05 April 2013

Get access

Summary

Professor Arnold is well known for his researches on a variety of topics in pure and applied mathematics, but perhaps no field owes more to him than singularity theory. In this volume are collected 7 survey articles of his on singularity theory that have appeared over the last decade. The first of these, written at a time (1968) when the subject was rapidly opening up, remains an excellent general introduction to the field as a whole.

However the core of the volume, consisting of 3 articles which appeared in 1973–75, consists of an account of the classification of critical points of smooth functions, and of the reinterpretations of a key class of functions (those with normal form depending on at most one parameter) – in relation to Lie groups, spherical and hyperbolic triangles, and definiteness of the intersection form – obtained by Arnold and his students during that period. Together, these results constitute one of the most beautiful discoveries in mathematics in recent years: and the further detailed study of these classes of singularities has revealed at each stage unexpected and rich structure.

Although Arnold does not shrink from describing the detailed calculations from which these lists are derived (the articles contain extensive – though not complete – sections of proofs) the surveys are far from being dry lists. He shows how a problem on estimating oscillatory integrals led him to start classifying functions; and by defining and computing an invariant (the ‘Arnold index’) applies the classification back to the original problem. In another paper, he relates the singularities of functions to those of projections of Lagrangian (and Legendre) submanifolds, and to the structure of caustics. The analysis of singularities of evolutes led to an extension of the work, in which singularities on the boundary of a manifold are investigated: an extension which in many ways completes the pattern set by the original.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×