تقارن هندسی

Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years--the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence. Chapters included in this book are: Chapter I, Introduction. The method of stationary phase; Appendix I, Morse's lemma and some generalizations; Chapter II, Differential operators and asymptotic solutions; Chapter III, Geometrical optics; Chapter IV, Symplectic geometry; Chapter V, Geometric quantization; Chapter VI, Geometric aspects of distribution; Appendix to Chapter VI, The Plancherel formula for the complex semisimple Lie groups; Chapter VII, Compound Asymptotics; Appendix II, Various functorial constructions; Index. surv14-frnt.pdf -1 Frontmatter 1 Title 1 Copyright page 2 Dedication 3 Preface 4 Notation 13 Contents 14 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chI.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase 16 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-appI.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations 31 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chII.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions 35 1. Differential operators 35 2. Asymptotic sections 41 3. The Luneburg-Lax-Ludwig technique 44 4. The methods of characteristics 48 5. Bicharacteristics 55 6. The transport equation 64 7. The Maslov cycle and the Bohr-Sommerfeld quantization conditions 72 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chIII.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical optics 85 1. The laws of refraction and reflection 85 2. Focusing and Magnification 92 3. Hamilton's method 97 4. First order optics 103 5. The Seidel aberrations 109 6. The asymptotic solution of Maxwell's equations 115 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chIV.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry 123 1. The Darboux-Weinstein theorem 123 2. Symplectic vector spaces 129 3. The cross index and the Maslov class 144 4. Functorial properties of Lagrangian submanifolds 163 5. Local parametrizations of Lagrangian submanifolds 167 6. Periodic Hamiltonian systems 182 7. Homogeneous symplectic spaces 194 8. Multisymplectic structures and the calculus of variations 218 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chV.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization 227 1. Curvature forms and vector bundles 227 2. The group of automorphisms of an Hermitian line bundle 235 3. Polarizations 242 4. Metalinear manifolds and half forms 265 5. Metaplectic manifolds 275 6. The pairing of half form sections 287 7. The metaplectic representation 290 8. Some examples 304 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chVI.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions 318 1. Elementary functorial properties of distributions 318 2. Traces and characters 329 3. The wave front set 337 4. Lagrangian distributions 355 5. The symbol calculus 367 Appendix to Section 5 376 6. Fourier integral operators 377 7. The transport equation 386 8. Some applications to spectral theory 392 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-app-to-VI.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups 401 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-chVII.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics 412 0. Introduction 412 1. The asymptotic Fourier transform 413 2. The frequency set 417 3. Functorial properties of compound asymptotics 422 4. The symbol calculus 427 5. Pointwise behavior of compound asymptotics and Bernstein's theorem 438 Appendix to Section 5 of Chapter VII 442 6. Behavior near caustics 447 7. Iterated S_1 and S{_2,0} singularities, computations 460 8. Proofs of the normal forms 469 9. Behavior near caustics (continued) 475 Appendix II. Various functorial constructions -1 Endmatter -1 surv14-appII.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions 482 1. The category of smooth vector bundles 482 2. The fiber product 485 Endmatter -1 surv14-bck.pdf 1 Frontmatter -1 Chapter I. Introduction. The method of stationary phase -1 Appendix I. Morse's lemma and some generalizations -1 Chapter II. Differential operators and asymptotic solutions -1 Chapter III. Geometrical options -1 Chapter IV. Symplectic geometry -1 Chapter V. Geometric quantization -1 Chapter VI. Geometric aspects of distributions -1 Appendix to Chapter VI. The Plancherel formula for the complex semi-simple Lie groups -1 Chapter VII. Compound asymptotics -1 Appendix II. Various functorial constructions -1 Endmatter 489 Index 489 Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought. This book intends to develop these themes, and present some of the advances, using the language of differential geometry as a unifying influence.