18.155 | Fall 2004 | Graduate

Differential Analysis

Calendar

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 

on the Line

 

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 VCc(Rn)

 
25

Questions

Trace as Integral of the Kernel over the Diagonal

Microlocal Analysis

 

Course Info

Departments
As Taught In
Fall 2004
Level
Learning Resource Types
Lecture Notes
Problem Sets with Solutions