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Introduction to the H-principle
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Introduction to the H-principle

Introduction to the H-principle

Yakov Eliashberg, N. Mishachev - Collection Graduate Studies in Mathematics

198 pages, parution le 08/08/2002

Résumé

One of the most powerful modern methods of solving partial differential equations is Gromov's h-principle. It has also been, traditionally, one of the most difficult to explain. This book is the first broadly accessible exposition of the principle and its applications. The essence of the h-principle is the reduction of problems involving partial differential relations to problems of a purely homotopy-theoretic nature. Two famous examples of the h-principle are the Nash-Kuiper $C1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology. Gromov transformed these examples into a powerful general method for proving the h-principle. Both of these examples and their explanations in terms of the h-principle are covered in detail in the book. The authors cover two main embodiments of the principle: holonomic approximation and convex integration. The first is a version of the method of continuous sheaves. The reader will find that, with a few notable exceptions, most instances of the h-principle can be treated by the methods considered here. There are, naturally, many connections to symplectic and contact geometry. The book would be an excellent text for a graduate course on modern methods for solving partial differential equations. Geometers and analysts will also find much value in this very readable exposition of an important and remarkable technique.

Sommaire

  • Intrigue
  • Thom transversality theorem
  • Holonomic approximation
  • Applications
  • Homotopy principle
  • Open Diff V-invariant differential relations
  • Applications to closed manifolds
  • Symplectic and contact structures on open manifolds
  • Symplectic and contact structures on closed manifolds
  • Embeddings into symplectic and contact manifolds
  • Microflexibility and holonomic mathcal{R}-approximation
  • First applications of microflexibility
  • Microflexible mathfrak{U}-invariant differential relations
  • Further applications to symplectic geometry
  • Homotopy principle for ample differential relations
  • Directed immersions and embeddings
  • First order linear differential operators
  • Nash-Kuiper theorem
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Caractéristiques techniques

  PAPIER
Éditeur(s) American Mathematical Society (AMS)
Auteur(s) Yakov Eliashberg, N. Mishachev
Collection Graduate Studies in Mathematics
Parution 08/08/2002
Nb. de pages 198
Format 18 x 26
Couverture Relié
Poids 620g
Intérieur Noir et Blanc
EAN13 9780821832271
ISBN13 978-0-821-83227-1

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