18.155, Differential Analysis I, Fall 2021

Instructor:Semyon Dyatlov
Class hours:TR 2:30–4 in 2-147
Office hours:Tue 1:30–2:30 in 2-377 and by appointment (possibly over Zoom)
TA:Rose Zhang, office hours Mondays 4–5 PM over Zoom (link in email)
Grading:Based on weekly problem sets, which will be due mostly on Tuesdays and collected in class on due dates. You may use TeX or you may write your solutions by hand; in the latter case please make sure they are easy to read! You may consult anyone and anything but you need to absorb and write out the solutions yourself. Anything close to copying is not permitted.
You might find Pset Partners useful for forming study groups.
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 briefly but familiarity with 18.101 will make things more comfortable. We may also use a bit of complex analysis (18.112) e.g. Cauchy integral formula.
Materials:There is no official textbook, I will provide lecture notes. But there are some books/notes you may find useful:

[FJ] Friedlander–Joshi: a short introduction to distribution theory and Sobolev spaces. I will use some of it in the first part of the course
[M] Lecture notes by Richard Melrose: I will use some of them for elliptic theory and other topics
[H] Hömander's Volume I: a classic introduction to distribution theory and Fourier transform. Too in-depth for our purposes but very good as a reference manual

Schedule (in reverse time order)

Thu Dec 9 TBA
Tue Dec 7 TBA
Thu Dec 2 TBA
Tue Nov 30 TBA
Tue Nov 23 TBA
Thu Nov 18 TBA
Tue Nov 16 TBA
Tue Nov 9 TBA
Thu Nov 4 TBA
Tue Nov 2 TBA
Thu Oct 28 TBA
Tue Oct 26 TBA
Thu Oct 21 TBA
Tue Oct 19 TBA
Thu Oct 14 TBA
Tue Oct 12 TBA
Thu Oct 7 TBA
Tue Oct 5 TBA
Thu Sep 30 TBA
Tue Sep 28 TBA
Thu Sep 23 TBA
Tue Sep 21 Differential operators acting on distributions.
Lecture notes §3
Thu Sep 16 Distributions: definition, convergence, localization (see notes §2).
Problemset 2, due Tue Sep 28
Tue Sep 14 Convolutions and mollifiers. Partitions of unity. A function is determined by its integrals against smooth compactly supported functions (see notes §1). Distributions.
Lecture notes §2
Suggested reading: [FJ, §§1.3–1.5] or [H, §§2.1–2.2]
Thu Sep 9 Prologue: motivation for distributions and elliptic regularity. Review: the spaces Lp, C etc. Riesz representation theorem for L2.
Lecture notes §1
Suggested reading: [FJ, §§1.1–1.2], [M, §§1.1–1.4, 2.1, 3.1], or [H, §§1.1–1.2, 1.3 up to Thm. 1.3.2, 1.4 skipping to Thm. 1.4.4]
Problemset 1, due Tue Sep 21

Last updated: Sep 16, 2021