| 1 |
Space of Continuous Functions, Dual Space, Positivity |
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| 2 |
Outer Measures and Measures |
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| 3 |
Caratheodory's Theorem |
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| 4 |
Measurable Functions and the Integral - Including Lebesgue's Theorem of Dominated Convergence |
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| 5 |
Riesz Representation, Lp Spaces and Completeness, L2(X,μ) and Hilbert Space |
Problem set 1 due |
| 6 |
Riesz Representation for Hilbert Space
Differentiability and Schwartz Space of Test Functions |
Problem set 2 due |
| 7 |
Properties of S(Rn)
Tempered Distributions
Differentiation and Differential Operators
Fourier Transform |
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| 8 |
Bump Functions
Characterization of δ Fourier Inversion
Plancherel Formula |
Problem set 3 due |
| 9 |
Convolution and Density
Fourier Transform on L2(Rn) |
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| 10 |
Sobolev Spaces and Sobolev Embedding
Duality between Sobolev Spaces |
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| 11 |
Schwartz Representation Theorem
Fundamental Solution of ∂x + i∂y
Support of a Distribution; Distributions of Compact Support (Start) |
Problem set 4 due |
| 12 |
Compact Supports
Convolution of Distributions supp(u*v) ⊂ supp(u)+ supp(v) if one, at least, has Compact Support
Fundamental Solutions |
Problem set 5 due |
| 13 |
Singular Support, Hypoellipticity, Ellipticity - Parametrices for Elliptic Operators |
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| 14 |
Fundamental Solution of the Heat Operator, Hypoellipticity, Initial Value Problem
Homogeneity
The Distributions xz±, z ∈ C\(-N) |
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| 15 |
Distributions Supported at 0
Homogeneous Distributions of Order  on the Line
Hadamard Regularization
Cone Supports |
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| 16 |
Singular Support and Products
Conic Support and Convolution |
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| 17 |
Wavefront Set Refines Singular Support
Scattering Wavefront Set
Product and Wavefront Set
The Wave Equation |
Problem set 6 due |
| 18 |
Fundamental Solution of the Wave Equation
Solution to the Cauchy Problem |
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| 19 |
Operators and Schwartz' Kernel Theorem |
Problem set 7 due |
| 20-24 |
Lidskii's Theorem
Δ on the Torus
Self-Adjointness of Δ+V
Spectral Decomposition
Wave Equation on the Torus
Wave Equation on Torus with Potential Δ+V on Rn with V ∈ Cc(Rn) |
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| 25 |
Questions
Trace as Integral of the Kernel over the Diagonal
Microlocal Analysis |
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