David Spring

Convex Integration Theory

Solutions to the h-principle in geometry and topology

• Broschiertes Buch

Produktbeschreibung

§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, adistinguishing 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.

Produktdetails

• Modern Birkhäuser Classics
• Verlag: Birkhäuser / Springer Basel / Springer, Basel
• Artikelnr. des Verlages: 978-3-0348-0059-4
• Seitenzahl: 213
• Englisch
• Abmessung: 13mm x 157mm x 236mm
• Gewicht: 344g
• ISBN-13: 9783034800594
• ISBN-10: 3034800592
• Artikelnr.: 32200111

Autorenporträt

David Spring is a Professor of mathematics at the Glendon College in Toronto, Canada.

Inhaltsangabe

Convex Hulls.- Analytic Theory.- Open Ample Relations in 1-Jet Spaces.- Microfibrations.- The Geometry of Jet Spaces.- Convex Hull Extensions.- Ample Relations.- Systems of Partial Differential Equations.- Relaxation Theory.