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Microlocal Analysis and Applications I MAT1063 Fall 2007

Description

It will be more introduction and theory than applications: Content

• Theory of Distributions.
  • Classes D,E,S and their dual D',E',S'. Basic operations, Fourier transform.
  • Sobolev spaces H^s on R^d.
  • Paley-Wiener theorem.
• Calculus of Pseudodifferential Operators:
  • Symbols, Quantization, Calculus.
  • Oscillatory Front Sets, coherent states, microlocalization.
  • Inverse of elliptic operator, resolvent.
  • Functional calculus.
• Analysis of Pseudodifferential Operators.
  • L²-estimates
  • Gårding inequalities.
• Pseudodifferential Operators and Boundary Value Problems.
  • Classical pseudodifferential operators.
  • Parametrix construction for elliptic boundary value problem.
  • Other types of operators appearing in parametrix construction.
  • Dirichlet-to-Neumann operator.
  • Non-elliptic boundary value problems.
• Applications to Hyperbolic Systems.
  • Proof of well-posedness of the Cauchy problem for strictly hyperbolic systems.

Dependencies

(They are not listed as pre-requisites)

• Real Analysis (graduate or undergraduate);
• Complex Analysis (graduate or undergraduate, or even non-specialist);
• Ordinary Differential Equations (graduate or undergraduate);
• Partial Differential Equations (graduate or undergraduate).

Follow-up: MAT1075 Microlocal Analysis and Applications II

Evaluation

Microlocal Analysis and Applications II MAT1075 Spring 2008

• Fourier Integral Operators:
  • Oscillatory solutions, relation to classical dynamics.
  • Elements of classical dynamics.
  • Lagrangian distributions, phase functions and amplitudes, Lagrangian manifolds.
  • Canonical graphs and Fourier Integral Operators.
  • Metaplectic operators.
  • Oscillatory solutions near and beyond caustics.
  • Maslov Canonical Operator.
• Propagation of singularities.
  • Fourier Integral Operators Approach.
  • Heisenberg approach.
  • Coherent States approach.
  • Energy estimates approach.
  • Propagation near boundary (survey)
• Applications to Spectral Asymptotics.
  • Tauberian Approach.
  • Poisson Relations, Spectrum and Length Spectrum.
  • Sharp spectral asymptotics.

Evaluation