| Instructor: | Semyon Dyatlov |
| Class hours: | TR 11–12:30 in 2-136 |
| Office hours: | Thu 10–11 in 2-377 and by appointment (possibly over Zoom) |
| TA: | Henry Zeng |
| TA office hours: | Mondays 2–3 PM in 2-390 |
| Canvas website: | Click here. Used to host pset solutions, exam problems and solutions, and pset and exam scores. |
| Grading: | The grade is determined based on the minimum of your total problem set score and the total exam score (both rescaled to have the same maximal possible number of points). Each of the 3 exams counts the same amount. The reason is: in the age of AI, I do not want to incentivize students to optimize their grade by maximizing their problem set score. But solving problem sets is an important part of learning and I do want to incentivize to make a good effort on these. |
| Problem sets: |
9 weekly problem sets, collected in class/on Canvas most Tuesdays (Sep 22, Sep 29, Oct 6, Oct 20, Oct 27, Nov 3, Nov 10, Nov 17, Dec 1).
You may use TeX or you may write your solutions by hand; in the latter case please make sure they are easy to read!
Your scores will be posted on Canvas and the papers will be returned in class or during office hours. You might find Pset Partners useful for forming study groups. |
| AI policy: | The goal of the problem sets is to make you work through the concepts in the course yourselves. Using AI to do the problem sets for you
robs you of an opportunity to learn and misses the point of education. On the other hand, there could be some value in using AI tools, e.g. for literature review, and it is impossible to enforce a strict no-AI policy on problem sets. So here is the policy for this semester: You can consult AI when working on problem sets, just like you would consult your classmates. However, you have to write the solutions yourselves and you have to understand what is in the solutions. If your solutions are typed, then you have to do the typing yourselves – no copy-pasting from AI, no asking it to prepare a file even if you edited it later etc. (You can ask AI to prepare a sample TeX file if you need to learn how to use TeX, and ask it for help with formatting, but you cannot make it prepare any of the actual 18.155 problem set content.) If I suspect that a student has used AI to write problem set solutions, I can ask them to come to office hours and ask them some questions about the problem set, and if the student is unable to answer, I can assign a score of 0 on that problem set (and I reserve the right to refer the student to COD). Same applies if a student got their hands on previous years' solutions and copy/pasted them (this is not really an AI problem). My sincere hope however is that you are eager to learn and I will not have to deal with many cases of academic dishonesty. |
| Exams: | Two in-person midterm exams during class time, on Oct 8 and Nov 12, and a final exam, on Dec 14, 9:00 AM–12:00 PM in 2-136. No outside assistance (e.g. devices or notes) allowed. |
| Description: | This course presents fundamental techniques for the rigorous study of partial differential equations (PDEs). We will start with the theory of distributions, Fourier transform, and Sobolev spaces, and explore some applications of these to constant coefficient PDEs such as Δu=f. We next go to more advanced applications as time permits, such as discreteness of the spectrum of the Laplacian on compact Riemannian manifolds and/or Hodge Theory. |
| Prerequisites: |
The official prerequisites are 18.102 or 18.103. The course will use basic concepts from functional analysis such as bounded operators on Banach spaces. It will also use Lebesgue integral as a black box. More complicated functional analysis will be reviewed as needed.
The latter part of the course will use manifolds – I will review them very briefly but familiarity with 18.101 will make things more comfortable.
We will also be using a bit of complex analysis (18.112) but it is not a hard prerequisite. I will check prerequisites for undergraduate students. Those who do not satisfy those and want to continue being enrolled in the class will have to convince me that they know the required material, for example by coming to my office and answering several questions on the relevant topics. |
| Lecture notes: | Click here. |
| Thu | Dec 10 |
TBA Reading: TBA |
| Tue | Dec 8 |
TBA Reading: TBA |
| Thu | Dec 3 |
TBA Reading: TBA |
| Tue | Dec 1 |
TBA Reading: TBA Pset 9 due |
| Tue | Nov 24 |
TBA Reading: TBA |
| Thu | Nov 19 |
TBA Reading: TBA |
| Tue | Nov 17 |
TBA Reading: TBA Pset 8 due |
| Thu | Nov 12 | Midterm 2 |
| Tue | Nov 10 |
TBA Reading: TBA Pset 7 due |
| Thu | Nov 5 |
TBA Reading: TBA |
| Tue | Nov 3 |
TBA Reading: TBA Pset 6 due |
| Thu | Oct 29 |
TBA Reading: TBA |
| Tue | Oct 27 |
TBA Reading: TBA Pset 5 due |
| Thu | Oct 22 |
TBA Reading: TBA |
| Tue | Oct 20 |
TBA Reading: TBA Pset 4 due: Exercises 6.1, 6.2, 7.1, 7.3, 7.5, 7.6, 8.1. |
| Thu | Oct 15 |
TBA Reading: §§7.3–8.3. |
| Thu | Oct 8 | Midterm 1 |
| Tue | Oct 6 |
Tensor product of distributions. Distributional kernels and the Schwartz kernel theorem (only sketch for the proof of existence). Adjoints of operators (started). Reading: §§7.1–7.3 Pset 3 due: Exercises 4.1, 4.4, 4.5, 5.1, 5.2, 5.3. |
| Thu | Oct 1 |
Homogeneous distributions on the real line and solving the equation xu=1.
Convolution of a distribution with a test function. Approximation of distributions
by test functions. Reading: §§5.2.2–6.2. |
| Tue | Sep 29 |
Distributions supported at one point (the proof). Homogeneous distributions.
Extending homogeneous distributions through the origin. Homogeneous distributions
on the real line. Reading: §§4.4–5.2.1. Pset 2 due: Exercises 2.4, 2.5, 3.1, 3.2, 3.5, 3.6. |
| Thu | Sep 24 |
Distributions with compact support as dual to the space of smooth functions.
Fréchet metric and Banach–Steinhaus for distributions.
Distributions supported at one point. Reading: §§4.2–4.4. |
| Tue | Sep 22 |
Multiplying distributions by smooth functions. Solving the differential equations
∂xu=0 and xu=0. Support of a distribution. Reading: §§3.1.3–4.1. Pset 1 due: Exercises 1.1, 1.2, 1.3, 2.1, 2.2. |
| Thu | Sep 17 |
Convergence of test functions and distributions. Localization and the sheaf property of distributions. Differentiating distributions. Reading: §§2.2–3.1. |
| Tue | Sep 15 |
The spaces C∞. Convolution and aproximation by smooth functions. Partitions of unity and recovering a function from its integrals against
test functions. Definition of distributions and embedding functions into the space of distributions. Reading: §§1.2.3–2.1. |
| Thu | Sep 10 |
Prologue: motivation for distributions and elliptic regularity. Review: the spaces Lp and Ck. Reading: §§1.1–1.2.3. |