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Convex Integration Theory: Solutions to the h-principle in geometry and topology: Monographs in Mathematics, cartea 92

Editat de David Spring
en Limba Engleză Hardback – 18 dec 1997
§1. Historical Remarks Convex Integration theory, first 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 classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes­ sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse­ 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 Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par­ tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods.
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Specificații

ISBN-13: 9783764358051
ISBN-10: 376435805X
Pagini: 228
Ilustrații: VIII, 213 p. 2 illus.
Dimensiuni: 155 x 235 x 18 mm
Greutate: 0.56 kg
Ediția:1998
Editura: Birkhäuser Basel
Colecția Birkhäuser
Seria Monographs in Mathematics

Locul publicării:Basel, Switzerland

Public țintă

Research

Cuprins

1 Introduction.- §1 Historical Remarks.- §2 Background Material.- §3 h-Principles.- §4 The Approximation Problem.- 2 Convex Hulls.- §1 Contractible Spaces of Surrounding Loops.- §2 C-Structures for Relations in Affine Bundles.- §3 The Integral Representation Theorem.- 3 Analytic Theory.- §1 The One-Dimensional Theorem.- §2 The C?-Approximation Theorem.- 4 Open Ample Relations in Spaces of 1-Jets.- §1 C°-Dense h-Principle.- §2 Examples.- 5 Microfibrations.- §1 Introduction.- §2 C-Structures for Relations over Affine Bundles.- §3 The C?-Approximation Theorem.- 6 The Geometry of Jet spaces.- §1 The Manifold X?.- §2 Principal Decompositions in Jet Spaces.- 7 Convex Hull Extensions.- §1 The Microfibration Property.- §2 The h-Stability Theorem.- 8 Ample Relations.- §1 Short Sections.- §2 h-Principle for Ample Relations.- §3 Examples.- §4 Relative h-Principles.- 9 Systems of Partial Differential Equations.- §1 Underdetermined Systems.- §2 Triangular Systems.- §3 C1-Isometric Immersions.- 10 Relaxation Theorem.- §1 Filippov’s Relaxation Theorem.- §2 C?-Relaxation Theorem.- References.- Index of Notation.

Recenzii

"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

Notă biografică

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

Textul de pe ultima copertă

This book provides a comprehensive study of convex integration theory in immersion-theoretic topology. Convex integration theory, developed originally by M. Gromov, provides general topological methods for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDE theory and to optimal control theory. Though topological in nature, the theory is based on a precise analytical approximation result for higher order derivatives of functions, proved by M. Gromov. This book is the first to present an exacting record and exposition of all of the basic concepts and technical results of convex integration theory in higher order jet spaces, including the theory of iterated convex hull extensions and the theory of relative h-principles. A second feature of the book is its detailed presentation of applications of the general theory to topics in symplectic topology, divergence free vector fields on 3-manifolds, isometric immersions, totally real embeddings, underdetermined non-linear systems of PDEs, the relaxation theorem in optimal control theory, as well as applications to the traditional immersion-theoretical topics such as immersions, submersions, k-mersions and free maps.
 
The book should prove useful to graduate students and to researchers in topology, PDE theory and optimal control theory who wish to understand the h-principle and how it can be applied to solve problems in their respective disciplines.
 
------  Reviews
 
The first eight chapters of Spring’s monograph contain a detailed exposition of convex integration theory for open and ample relations with detailed proofs that were often omitted in Gromov’s book. (…) Spring’s bookmakes 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
 
This volume provides a comprehensive study of convex integration theory. (…) We recommended the book warmly to all interested in differential topology, symplectic topology and optimal control theory.
- Matematica

Caracteristici

Comprehensive and systematic monograph on convex integration theory Indispensable to all interested in differential topology, symplectic topology and optimal control theory Addresses as well as researchers Includes supplementary material: sn.pub/extras