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Lec #TOPICSKEY DATES
1Space of Continuous Functions, Dual Space, Positivity
2Outer Measures and Measures
3Caratheodory's Theorem
4Measurable Functions and the Integral - Including Lebesgue's Theorem of Dominated Convergence
5Riesz Representation, Lp Spaces and Completeness, L2(X,μ) and Hilbert SpaceProblem set 1 due
6Riesz Representation for Hilbert Space

Differentiability and Schwartz Space of Test Functions
Problem set 2 due
7Properties of S(Rn)

Tempered Distributions

Differentiation and Differential Operators

Fourier Transform
8Bump Functions

Characterization of δ Fourier Inversion

Plancherel Formula
Problem set 3 due
9Convolution and Density

Fourier Transform on L2(Rn)
10Sobolev Spaces and Sobolev Embedding

Duality between Sobolev Spaces
11Schwartz Representation Theorem

Fundamental Solution of ∂x + i∂y

Support of a Distribution; Distributions of Compact Support (Start)
Problem set 4 due
12Compact Supports

Convolution of Distributions supp(u*v) ⊂ supp(u)+ supp(v) if one, at least, has Compact Support

Fundamental Solutions
Problem set 5 due
13Singular Support, Hypoellipticity, Ellipticity - Parametrices for Elliptic Operators
14Fundamental Solution of the Heat Operator, Hypoellipticity, Initial Value Problem

Homogeneity

The Distributions xz±, z ∈ C\(-N)
15Distributions Supported at 0

Homogeneous Distributions of Order Homogeneous distributions. on the Line

Hadamard Regularization

Cone Supports
16Singular Support and Products

Conic Support and Convolution
17Wavefront Set Refines Singular Support

Scattering Wavefront Set

Product and Wavefront Set

The Wave Equation
Problem set 6 due
18Fundamental Solution of the Wave Equation

Solution to the Cauchy Problem
19Operators and Schwartz' Kernel TheoremProblem set 7 due
20-24Lidskii'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 VCc(Rn)
25Questions

Trace as Integral of the Kernel over the Diagonal

Microlocal Analysis

 








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