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Introduction to Microlocal Analysis by Richard Melrose


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Introduction to Microlocal Analysis written by Richard Melrose, Massachusetts Institute of Technology . This is an other book of mathematics cover the following topics.

  • 1. Tempered distributions and the Fourier transform
    1.1. Schwartz test functions
    1.2. Linear transformations
    1.3. Tempered distributions
    1.4. Two big theorems
    1.5. Examples
    1.6. Two little lemmas
    1.7. Fourier transform
    1.8. Differential operators
    1.9. Radial compactification
    1.10. Problems

  • 2. Pseudodifferential operators on Euclidean space
    2.1. Symbols
    2.2. Pseudodifferential operators
    2.3. Composition
    2.4. Reduction
    2.5. Asymptotic summation
    2.6. Residual terms
    2.7. Proof of Composition Theorem
    2.8. Quantization and symbols
    2.9. Principal symbol
    2.10. Ellipticity
    2.11. Elliptic regularity
    2.12. The Laplacian
    2.13. L2 boundedness
    2.14. Square root
    2.15. Proof of Boundedness
    2.16. Sobolev boundedness
    2.17. Consequences
    2.18. Polyhomogeneity
    2.19. Linear invariance
    2.20. Problems

  • 3. Isotropic and scattering calculi
    3.1. Isotropic operators
    3.2. Scattering operators
    3.3. The residual algebra isotropic algebra
    3.4. The residual isotropic ring
    3.5. Exponential and logarithm
    3.6. Fredholm property
    3.7. The harmonic oscillator
    3.8. L2 boundedness and compactness
    3.9. Sobolev spaces
    3.10. The residual group
    3.11. Representations
    3.12. Symplectic invariance of the isotropic product
    3.13. Complex order
    3.14. Traces on the residual algebra
    3.15. Fredholm determinant
    3.16. Fredholm alternative
    3.17. Resolvent and spectrum
    3.18. Residue trace
    3.19. Exterior derivation
    3.20. Regularized trace
    3.21. Projections
    3.22. Complex powers
    3.23. Index and invertibility
    3.24. Variation 1-form
    3.25. Determinant bundle
    3.26. Index bundle
    3.27. Index formulæ
    3.28. Isotropic essential support
    3.29. Isotropic wavefront set
    3.30. Isotropic FBI transform
    3.31. Problems 95

  • 4. Microlocalization
    4.1. Calculus of supports
    4.2. Singular supports
    4.3. Pseudolocality
    4.4. Coordinate invariance
    4.5. Problems
    4.6. Characteristic variety
    4.7. Wavefront set
    4.8. Essential support
    4.9. Microlocal parametrices
    4.10. Microlocality
    4.11. Explicit formulations
    4.12. Wavefront set of KA
    4.13. Elementary calculus of wavefront sets
    4.14. Pairing
    4.15. Multiplication of distributions
    4.16. Projection
    4.17. Restriction
    4.18. Exterior product
    4.19. Diffeomorphisms
    4.20. Products
    4.21. Pull-back
    4.22. The operation F*
    4.23. Wavefront relation
    4.24. Applications
    4.25. Problems

  • 5. Pseudodifferential operators on manifolds
    5.1. C8 structures
    5.2. Form bundles
    5.3. Pseudodifferential operators
    5.4. Pseudodifferential operators on vector bundles
    5.5. Laplacian on forms
    5.6. Hodge theorem
    5.7. Pseudodifferential projections
    5.8. Heat kernel
    5.9. Resolvent
    5.10. Complex powers
    5.11. Problems

  • 6. Elliptic boundary problems
    6.1. Manifolds with boundary
    6.2. Smooth functions
    6.3. Distributions
    6.4. Boundary Terms
    6.5. Sobolev spaces
    6.6. Dividing hypersurfaces
    6.7. Rational symbols
    6.8. Proofs of Proposition
    6.9. Inverses
    6.10. Smoothing operators
    6.11. Left and right parametrices
    6.12. Right inverse
    6.13. Boundary map
    6.14. Calder`on projector
    6.15. Poisson operator
    6.16. Unique continuation
    6.17. Boundary regularity
    6.18. Pseudodifferential boundary conditions
    6.19. Gluing
    6.20. Local boundary conditions
    6.21. Absolute and relative Hodge cohomology
    6.22. Transmission condition

  • 7. Scattering calculus
  • 8. The wave kernel
    8.1. Hamilton-Jacobi theory
    8.2. Riemann metrics and quantization
    8.3. Transport equation
    8.4. Problems
    8.5. The wave equation
    8.6. Forward fundamental solution
    8.7. Operations on conormal distributions
    8.8. Weyl asymptotics
    8.9. Problems

  • 9. K-theory
    9.1. Vector bundles
    9.2. The ring K(X)
    9.3. Chern-Weil theory and the Chern character
    9.4. K1(X) and the odd Chern character
    9.5. C* algebras
    9.6. K-theory of an algeba 199
    9.7. The norm closure of ?0(X)
    9.8. The index map
    9.9. Problems

  • 10. Hochschild homology
    10.1. Formal Hochschild homology
    10.2. Hochschild homology of polynomial algebras
    10.3. Hochschild homology of C8(X)
    10.4. Commutative formal symbol algebra
    10.5. Hochschild chains
    10.6. Semi-classical limit and spectral sequence
    10.7. The E2 term 10.8. Degeneration and convergence 10.9. Explicit cohomology maps 10.10. Hochschild holomology of ?-8(X) 10.11. Hochschild holomology of ?Z(X) 10.12. Morita equivalence 218

  • 11. The index formula
  • Appendix A. Bounded operators on Hilbert space

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