Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology (Modern Birkhäuser Classics)

Spring, David

ISBN 10: 3034800592 / ISBN 13: 9783034800594

Published by Birkhäuser, 2010

About this Item

This is an ex-library book and may have the usual library/used-book markings inside.This book has soft covers. In good all round condition. Please note the Image in this listing is a stock photo and may not match the covers of the actual item,450grams, ISBN:9783034800594. Seller Inventory # 5832913

Bibliographic Details

Title: Convex Integration Theory: Solutions to the ...

Publisher: Birkhäuser

Publication Date: 2010

Binding: Soft cover

Condition: Good

About this title

Synopsis:

1. Historical Remarks Convex Integration theory, ?rst introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.

Review:

"Spring's book makes no attempt to include all topics from convex integration theory or to uncover all of the gems in Gromov's fundamental account, but it will nonetheless (or precisely for that reason) take its place as a standard reference for the theory next to Gromov's towering monograph and should prove indispensable for anyone wishing to learn about the theory in a more systematic way."

--- Mathematical Reviews