Course Schedule
MATH 7540 - Spring 2025
Below is a course schedule. It contains an outline of course topics, homework, quiz and exam dates.
Note: This schedule is approximate and will evolve as the course progresses. Lectures may be rearranged and split between days depending on course progress. The exams dates are tentative and will be finalized at least a couple weeks before the date.
| Week | Date | Topics | Section | HW |
|---|---|---|---|---|
| 1 | Functional Analysis Preliminaries | Appendix D, Notes | ||
| M 1/13 | Introduction, Review of Measure Theory/Real Analysis | Appendix D, Notes | ||
| W 1/15 | More Real Analysis Preliminaries | Appendix D, Notes | ||
| F 1/17 | Banach and Hilbert Spaces | Appendix D, Notes | ||
| 2 | M 1/20 | MLK Jr. Holiday - No Class | ||
| W 1/22 | Classes Cancelled (Snow) | |||
| F 1/24 | Bounded Linear Operators, Dual Spaces | Appendix D, Notes | ||
| 3 | M 1/27 | Weak and Weak-* Topologies, Banach Alaoglu, Dunford Pettis | Appendix D, Notes | |
| W 1/29 | Compactness in Banach Spaces, Arzela-Ascoli | Appendix D, Notes | ||
| F 1/31 | Compactness in Banach Spaces, Riesz-Kolmogorov | Appendix D, Notes | ||
| 4 | Fourier Analysis on $S(\mathbb{R}^n)$, Extension to $L^p$, and Tempered Distributions | Notes | ||
| M 2/3 | The Fourier Transform on $S(\mathbb{R}^n)$ | Notes | ||
| W 2/5 | Properties of the Fourier Transform, Fourier Inversion | Notes | ||
| F 2/7 | Riemann-Lebesgue Lemma, Extension to $L^p$, Hausdorff-Young | Notes | ||
| 5 | M 2/10 | Tempered Distributions and the Fourier Transform on $S'(\mathbb{R}^n)$, Periodic Fourier Series | Notes | |
| W 2/12 | Clifford Lectures | |||
| F 2/14 | Clifford Lectures | |||
| 6 | Sobolev Spaces on $\mathbb{R}^n$ | Notes | ||
| M 2/17 | Sobolev Spaces on $\mathbb{R}^n$: Definitions via Fourier Transform, Basic Properties | Notes | ||
| W 2/19 | Sobolev Embedding Theorems ($H^s$) | Notes | ||
| F 2/21 | Compactness Theorems ($H^s$) | Notes | ||
| 7 | Singular Integrals (Calderón-Zygmund Theory) | Notes | ||
| M 2/24 | Intro to Singular Integrals: Riesz Potential | Notes | ||
| W 2/26 | Hardy-Littlewood-Sobolev Theorem | Notes | ||
| F 2/28 | Maximal Function, Vitali Covering Lemma | Notes | ||
| - | M 3/3 | Mardi Gras/Spring Break - No Class | ||
| W 3/5 | Mardi Gras/Spring Break - No Class | |||
| F 3/7 | Mardi Gras/Spring Break - No Class | |||
| 8 | M 3/10 | Proof of HLS, Fractional Sobolev Embeddings | Notes | |
| W 3/12 | Calderon-Zygmund Potentials, CZ Decomposition | Notes | ||
| F 3/14 | Proof of CZ Theorem (Weak Type Estimate) | Notes | ||
| 9 | M 3/17 | Proof of CZ Theorem ($L^p$ Estimate) | Notes | |
| Sobolev Spaces on Bounded Domains | Ch 5 | |||
| W 3/19 | Sobolev Spaces on Bounded Domains (Evans Ch. 5) | Ch 5 | ||
| F 3/21 | Extension Theorems (Bounded Domains) | Ch 5 | ||
| 10 | M 3/24 | Trace Theorems (Bounded Domains) | Ch 5 | |
| W 3/26 | Sobolev Embedding Theorems (Bounded Domains) | Ch 5 | ||
| F 3/28 | Compactness Theorems (Bounded Domains) | Ch 5 | ||
| 11 | Applications to Elliptic PDE | Ch 6 | ||
| M 3/31 | Applications to Elliptic PDE: Weak Solutions, Lax-Milgram | Ch 6 | ||
| W 4/2 | Applications to Elliptic PDE: Weak Solutions, Lax-Milgram (Continued) | Ch 6 | ||
| F 4/4 | Regularity Theory for Elliptic Equations | Ch 6 | ||
| 12 | M 4/7 | Regularity Theory Continued | Ch 6 | |
| W 4/9 | Maximum Principles for Elliptic Equations | Ch 6 | ||
| F 4/11 | Maximum Principles Continued | Ch 6 | ||
| 13 | M 4/14 | Lagniappe Days (No Class) | ||
| W 4/16 | Lagniappe Days (No Class) | |||
| F 4/18 | Eigenvalues and Eigenfunctions | Ch 6 | ||
| 14 | Semigroup Theory | Ch 7, Notes | ||
| M 4/21 | Semigroup Theory: Definitions, Examples | Ch 7, Notes | ||
| W 4/23 | Hille-Yosida Theorem | Ch 7, Notes | ||
| F 4/25 | Applications of Semigroup Theory | Ch 7, Notes | ||
| 15 | M 4/28 | Applications of Semigroup Theory Continued | Ch 7, Notes | |
| Advanced Topic | Notes/Papers | |||
| W 4/30 | Advanced Topic (e.g., DeGiorgi-Nash-Moser) | Notes/Papers | ||
- 2025