MTH943 Course Videos

I will post ≈30 minutes worth of video lectures every weekday between January 20 and April 16, with the exception of the March 2 & 3 university break days. Each individual video will be no more than 10 minutes in length; there will be multiple "short lectures" posted each day. It is entirely up to you whether you watch the videos each day or binge them once a week.

Week 1 (Jan 20 - Jan 26)

Week 2 (Jan 27 - Feb 2)

III.4 Classical versus Quantum (1/27)
III.5 Fourier Transform: Definition and Basic Properties (1/27)
III.6 Fourier Transform: Uniform Continuity (1/27)
III.7 Fourier Transform: Derivatives (1/28)
III.8 Schwartz Space (1/28)
III.9 Properties of Gaussians (1/28)
III.10 Convolution: Young's Inequality (1/29)
III.11 Convolution: Arithmetic and Differentiability (1/29)
III.12 Smooth Approximation of $L^p$ functions (1/29)
III.13 Fourier Inversion Formula (1/29)
III.14 Back to Schroedinger (2/1)
III.15 Ditto Wave and Klein-Gordon (2/1)
III.16 Toward $L^2$ Theory (2/1)
III.17 Density Argument; Fourier in $L^2$ (2/1)
III.18 Uncertainty Principles (2/2)

Linear Schrödinger Equation

IV.1 The Two Methods (2/2)
IV.2 Galilean Boosts (2/2)
IV.3 Conservation of Mass and Energy (2/2)

Week 3 (Feb 3 - Feb 9)

IV.4 Global Sobolev Inequality (2/3)
IV.5 Dispersive Smoothing (2/3)
IV.6 Dyadic Localization (2/3)
IV.7 Dyadic Reassembly, the Space $Y^{s,q}$ (2/4)
IV.8 Intro to Interpolation Theory (2/4)
IV.9 Interpolation Functors (2/4)
IV.10 Notable Interpolation Pairs (2/5)
IV.11 Dispersive Decay Again (2/5)
IV.12 Toward Translation Invariance (2/5)
IV.13 The $L^1$-$L^\infty$ estimate (2/8)
IV.14 Interpolating Again (2/8)
IV.15 Abstract Strichartz: Overview (2/8)
IV.16 The $TT^\ast$ Argument (2/8)
IV.17 The Hardy-Littlewood-Sobolev Inequality (2/9)
IV.18 Fractional Integration (2/9)
IV.19 Abstract Strichartz: Proof Part I (2/9)
IV.20 Abstract Strichartz: Proof Part II (2/9)

Week 4 (Feb 10 - Feb 16)

IV.21 Concrete Strichartz: Preliminaries (2/10)
IV.22 Classical Strichartz for Schrödinger (2/10)
IV.23 Integrated Local Mass Decay I (2/10)
IV.24 Integrated Local Mass Decay II (2/11)

Nolinear Schrödinger Equations

V.1 Overview (2/11)
V.2 Why do we Care? (2/11)
V.3 Inhomogeneous Equations and Duhamel (2/11)
V.4 Picard-Lindelöf Revisited (2/12)
V.5 Picard-Lindelöf Part 2 (2/12)
V.6 Gröwall's Inequality (2/12)
V.7 Using Gröwall for PDEs (2/12)
V.8 Local to Global (2/15)
V.9 Strichartz with Inhomogeneity I (2/15)
V.10 Strichartz with Inhomogeneity II (2/15)
V.11 The Nonlinearities (2/16)
V.12 $L^2$ Numerology (2/16)
V.13 Critical versus Subcritical (2/16)

Week 5 (Feb 17 - Feb 23)

V.14 Other Scalings (2/17)
V.15 $L^2$ Subcritical Local Wellposedness (2/17)
V.16 $L^2$ Subcritical Local Wellposedness II (2/17)
V.17 $L^2$ Subcritical Local Wellposedness III (2/18)
V.18 Comments on the Proof (2/18)
V.19 Mass Conservation (2/18)
V.20 Subcritical Global Wellposedness (2/19)
V.21 $L^2$ Critical Local Existence (2/19)
V.22 $L^2$ Critical Local Existence II (2/19)
V.23 $L^2$ Critical Small Data Global Wellposedness (2/19)
V.24 Critical versus Subcritical (Reprise) (2/22)
V.25 $H^1$ Numerology I (2/22)
V.26 $H^1$ Numerology II (2/22)
V.27 $H^1$ Wellposedness Results (2/22)
V.28 Energy Conservation: Heuristics (2/23)
V.29 Cubic NLS: Outline (2/23)
V.30 Cubic NLS: Persistence of Regularity I (2/23)

Week 6 (Feb 24 - Mar 5)

Note: University break days on Mar 2 and Mar 3. I will also post no lecture for Monday Mar 1. So going forward Weeks will end on Fridays.

V.31 Cubic NLS: Persistence of Regularity II (2/24)
V.32 Cubic NLS: Conservation of Energy (2/24)
V.33 Cubic NLS: Defocusing Global Wellposedness (2/24)
V.34 Cubic NLS: Focusing Small-Data Global Wellposedness (2/24)
V.35 Glassey's Virial Identity (2/25)
V.36 Cubic NLS: Focusing Large Data Blow-up (2/25)

Linear Klein-Gordon and Wave

VI.1 Overview and Notation (2/25)
VI.2 Classical Energy Conservation (2/26)
VI.3 Pointwise Estimates (2/26)
VI.4 Energy Current (2/26)
VI.5 Positivity of Energy Integral (2/26)
VI.6 Space-like Hypersurfaces (3/4)
VI.7 Finite Speed of Propagation (3/4)
VI.8 Lorentz Boosts (3/4)
VI.9 Hyperboloids (3/5)
VI.10 Our Strategy (3/5)
VI.11 Hyperboloidal Geometry (3/5)
VI.12 Hyperboloidal Flux (3/5)

Week 7 (Mar 8 - Mar 12)

VI.13 Adapted Sobolev Spaces (3/8)
VI.14 Some Basic Inequalities (3/8)
VI.15 Adapted Sobolev Inequalities: $p \lt d$ case (3/8)
VI.16 Adapted Sobolev Inequalities: $p = d$ case (3/8)
VI.17 Adapted Sobolev Inequalities: $p \gt d$ case (3/9)
VI.18 $p\gt d$ case continued (3/9)
VI.19 Higher Order Sobolev Inequalities (3/9)
VI.20 Wave-type $L^2$ Morrey Estimates (3/9)
VI.21 Klein-Gordon-type $L^2$ Morrey Estimates (3/10)
VI.22 Dispersion inside the Light Cone (3/10)
VI.23 Dispersion inside the Light Cone II (3/10)
VI.24 Dispersion inside the Light Cone III (3/11)
VI.25 Dispersion inside the Light Cone IV (3/11)
VI.26 Anisotropy (3/11)
VI.27 The Story so Far (3/12)
VI.28 Data with Compact Support (3/12)
VI.29 Spatially Homogenized Dispersive Estimates (3/12)
VI.30 Spatially Homogenized Dispersive Estimates II (3/12)

Week 8 (Mar 15 - Mar 19)

Nonlinear Equations with Small, Compactly Supported Initial Data

VII.1 Intro to Part VII (3/15)
VII.2 The Equations (3/15)
VII.3 Energy Identity (3/15)
VII.4 Finite Speed of Propagation I (3/15)
VII.5 Finite Speed of Propagation II (3/16)
VII.6 Finite Speed of Propagation III (3/16)
VII.7 Solution Operator (3/16)
VII.8 Inhomogeneous Equations (3/17)
VII.9 Local Wellposedness I (3/17)
VII.10 Local Wellposedness II (3/17)
VII.11 Local Wellposedness III (3/18)
VII.12 Spatially Localized Local Wellposedness (3/18)
VII.13 Nonlocalized Data Continued (3/18)
VII.14 Local Uniqueness (3/19)
VII.15 Small Data Global Wellposedness I (3/19)
VII.16 Small Data Global Wellposedness II (3/19)

Week 9 (Mar 22 - Mar 26)

Note: this week I am using a long-form video format. This is because the technical parts of the proof of small data global wellposedness is a little bit complicated, with many moving parts. Splitting things up into short videos makes cross referencing more complicated and the proof harder to follow.

You can find a copy of the handwritten notes, pre-annotation, HERE. I encourage having a copy open to follow along when watching the videos.