| Lec # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Space of Continuous Functions, Dual Space, Positivity | |
| 2 | Outer Measures and Measures | |
| 3 | Caratheodory's Theorem | |
| 4 | Measurable Functions and the Integral - Including Lebesgue's Theorem of Dominated Convergence | |
| 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 |
|
| 8 |
Bump Functions Characterization of δ Fourier Inversion Plancherel Formula |
Problem set 3 due |
| 9 |
Convolution and Density Fourier Transform on L2(Rn) |
|
| 10 |
Sobolev Spaces and Sobolev Embedding Duality between Sobolev Spaces |
|
| 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 | |
| 14 |
Fundamental Solution of the Heat Operator, Hypoellipticity, Initial Value Problem Homogeneity The Distributions xz±, z ∈ C\(-N) |
|
| 15 |
Distributions Supported at 0 Homogeneous Distributions of Order Hadamard Regularization Cone Supports |
|
| 16 |
Singular Support and Products Conic Support and Convolution |
|
| 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 |
|
| 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) |
|
| 25 |
Questions Trace as Integral of the Kernel over the Diagonal Microlocal Analysis |
on the Line