18.155, Differential Analysis I, Fall 2022

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)
Grading:Based on weekly problem sets, which will be due mostly on Thursdays (Sep 22, Sep 29, Oct 6, Oct 20, Oct 27, Nov 3, Nov 10, Nov 17, Dec 1, Dec 13) 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. 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.
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 will also be using a bit of complex analysis (18.112).
Previously:I taught this course in Fall 2021. I am planning on following roughly the same path.
Lecture notes:Click here. Frequently updated during the semester.

Schedule (in reverse time order)

Tue Dec 13 TBA
Thu Dec 8 TBA
Tue Dec 6 TBA
Thu Dec 1 TBA
Tue Nov 29 TBA
Tue Nov 22 TBA
Thu Nov 17 TBA
Tue Nov 15 TBA
Thu Nov 10 TBA
Tue Nov 8 TBA
Thu Nov 3 TBA
Tue Nov 1 TBA
Thu Oct 27 TBA
Tue Oct 25 TBA
Thu Oct 20 TBA
Tue Oct 18 TBA
Thu Oct 13 TBA
Thu Oct 6 TBA
Tue Oct 4 TBA
Thu Sep 29 TBA
Tue Sep 27 TBA
Thu Sep 22 TBA
Tue Sep 20 TBA
Thu Sep 15 TBA
Tue Sep 13 TBA
Thu Sep 8 TBA

Last updated: Jul 17, 2022