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 on Canvas)
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. 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. For a list of concepts from complex analysis (18.112) we will use, click here.
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] Hörmander'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 Differential forms. Hodge decomposition. Poincaré duality (see notes §18).
Tue Dec 7 Overview of various results on Laplacian spectrum: Weyl Law, nodal sets, lower bounds on eigenfunctions (see notes §17). Elliptic differential operators on vector bundles.
Lecture notes §18
Thu Dec 2 End of the proof of Fredholm property of elliptic operators (see notes §16). Spectral theorem for self-adjoint elliptic operators.
Lecture notes §17
If you want to learn more on the topics discussed in §17.2, you can try to look at lecture notes/reviews by Canzani, Zelditch, Logunov–Malinnikova, and/or myself
Tue Nov 30 Compact embeddings of Sobolev spaces continued. Review of Fredholm operators. Elliptic operators on compact manifolds are Fredholm. (See notes §16.)
Problemset 11, due Tue Dec 7
Tue Nov 23 Boundedness of pseudodifferential operators on Sobolev spaces. Elliptic estimates. Compact embeddings of Sobolev spaces.
Lecture notes §16
Suggested reading: [M, §§6.1–6.3]
Thu Nov 18 Elliptic parametrix and proof of Elliptic Regularity III; Borel's Theorem (see notes §15).
Problemset 10, due Tue Nov 30
Tue Nov 16 Pseudodifferential operators. Elliptic parametrix (see notes §15).
Tue Nov 9 Principal symbol and pullbacks. Differential operators on manifolds. Principal symbol as a function on the cotangent bundle (see notes §14). Elliptic differential operators. Statement of Elliptic Regularity III. Kohn–Nirenberg symbols.
Lecture notes §15, final version
Suggested reading: [M, §4.3]
Thu Nov 4 Distributions and Sobolev spaces on manifolds (see notes §13). Differential operators on Euclidean space. Example: the Laplace–Beltrami operator.
Lecture notes §14
Problemset 9, due Thu Nov 18
Tue Nov 2 Manifolds, tangent and cotangent bundles, Riemannian metrics.
Lecture notes §13
Thu Oct 28 Sobolev spaces: multiplication by cutoff functions, a double integral characterization. Elliptic regularity II (see notes §12).
Problemset 8, due Tue Nov 9
Tue Oct 26 Properties of Fourier transform on tempered distributions; Fourier transform of compactly supported distributions (see notes §11). Sobolev spaces: basic properties.
Lecture notes §12
Suggested reading: [FJ, §§9.1–9.3,8.6] or [H, §7.9 (first 2 pages),Thm.7.1.22]
Thu Oct 21 Fourier transform on Schwartz functions and tempered distributions. Fourier inversion formula.
Lecture notes §11
Suggested reading: [FJ, §§8.1–8.4] or [H, §7.1 (up to Thm. 7.1.15)]
Problemset 7, due Tue Nov 2
Tue Oct 19 Fundamental solution to the wave equation and applications (see notes §10).
Suggested reading: [FJ, §7.3] or [H, §6.2]
Thu Oct 14 Pullbacks by diffeomorphisms. Pullbacks by submersions.
Lecture notes §10
Suggested reading: [FJ, §§7.1–7.2] or [H, §6.1]
Problemset 6, due Tue Oct 26
Tue Oct 12 Fundamental solutions to constant coefficient PDEs, basic properties. Examples of fundamental solutions: Laplacian, the wave operator in 1+1 dimension. Singular support. Elliptic regularity I.
Lecture notes §9
Suggested reading: [FJ, §5.4] or [H,§§3.3(up to Thm. 3.3.2),4.4(up to Thm. 4.4.1)]
Thu Oct 7 Defining operators on distributions by duality (see notes §7). Convolution of distributions with compact support. Convolution of distributions whose supports sum properly.
Lecture notes §8
Suggested reading: [FJ, §§5.1–5.3] or [H, §4.2]
Problemset 5, due Tue Oct 19
Tue Oct 5 Tensor product of distributions. Operators between spaces of distributions and the Schwartz kernel theorem.
Lecture notes §7
Suggested reading: [FJ, §§4.1,4.3,6.1–6.2] or [H, Thm.2.1.3 and §§5.1–5.2]
Thu Sep 30 Convolution of a distribution and a smooth function, basic properties. Approximation of distributions by smooth functions with compact support.
Lecture notes §6
Suggested reading: [FJ, §5.2] (a bit awkward because we haven't defined convolution of distributions yet) or [H, §4.1 up to Thm. 4.1.5]
Problemset 4, due Tue Oct 12
Tue Sep 28 Homogeneous distributions: definition and basic homogeneous distributions on the real line. Banach–Steinhaus for distributions (see notes §6).
Lecture notes §5
Suggested reading: [FJ, §2.3] or [H, §3.2 up to the statement of Thm. 3.2.3]
Thu Sep 23 Support of a distribution. Distributions with compact support as the space dual to smooth functions. Distributions supported at a single point.
Lecture notes §4
Suggested reading: [FJ, &3] or [H, §2.3 up to Thm. 2.3.4]
Problemset 3, due Tue Oct 5
Tue Sep 21 Differentiation of distributions and multiplication by smooth functions. Solving basic ODEs: u'=0 and xu=0.
Lecture notes §3
Suggested reading: [FJ, §§2.1–2.2,2.4–2.5,2.7] or [H, §3.1 up to Thm. 3.1.7]
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: Dec 7, 2021