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)
TA:Zhenhao Li, office hours Wednesdays 2–3PM in 2-239A
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. The exercises are given in the lecture notes at the end of each chapter. The exercises with a star next to their numbers are optional. 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.

Schedule (in reverse time order)

Tue Dec 13 Degree theory. Overview of advanced results on Laplacian eigenfunctions.
Reading: §§17.3.6,16.2.
Thu Dec 8 Dirac operator and its ellipticity. Hodge Laplacian. Harmonic forms on compact Riemannian manifolds. Hodge decomposition and de Rham cohomology. Applications: Poincaré duality, index of the Dirac operator.
Reading: §§17.2, 17.3.3–17.3.5.
Tue Dec 6 A brief review of differential forms: exterior powers of the tangent bundle, wedge product, exterior derivative, integration, and Stokes's Theorem. Inner product on differential forms, Hodge star operator, codifferential operator.
Reading: §§17.1, 17.3.1–17.3.2.
Thu Dec 1 A touch of index theory. Spectral theory for self-adjoint elliptic operators on compact manifolds.
Reading: §§15.3.3, 16.1.
Tue Nov 29 Fredholm property of elliptic differential operators on compact manifolds. A bit of general theory of compact and Fredholm operators.
Reading: §§15.3.1–15.3.2.
Pset 10, due Tue Dec 13: Exercises 15.4, 15.5, 15.6*, 16.1, 17.1*, 17.2*.
Tue Nov 22 Continuity of pseudodifferential operators on Sobolev spaces. Elliptic estimate. Compact embedding for Sobolev spaces.
Reading: §§15.1–15.2.
Thu Nov 17 Elliptic parametrix and the proof of Elliptic Regularity III. Elliptic operators on vector bundles. Schur's bound.
Reading: §§14.2.1–14.2.3, 15.1.1.
Tue Nov 15 Proof of Borel's Theorem. Pseudodifferential operators. Composition of a differential operator and a pseudodifferential operator.
Reading: §§14.1.2–14.2.1.
Pset 9, due Thu Dec 1: Exercises 14.1, 14.2*, 14.3, 14.4, 14.5, 15.1*, 15.2, 15.3.
Thu Nov 10 Differential operators on manifolds and vector bundles. Statement of Elliptic Regularity III. Kohn–Nirenberg symbols. Asymptotic expansions and Borel's Theorem.
Reading: §§13.3,14.1.1–14.1.2.
Tue Nov 8 Vector bundles. Distributions and Sobolev spaces on manifolds. Differential operators.
Reading: §§13.1.8–13.3.1.
Pset 8, due Thu Nov 17: Exercises 13.1, 13.2, 13.3, 13.4, 13.5, 13.6*, 13.7, 13.8.
Thu Nov 3 Kohn–Nirenberg symbols. Proof of Elliptic Regularity II. Review of manifolds, tangent and cotangent bundles, Riemannian metrics, and integration.
Reading: §§12.2.3–12.2.4, 13.1.1–13.1.7.
Tue Nov 1 Characterization of Sobolev spaces in Fourier-less terms. Multiplication by a Schwartz function is a bounded operator on a Sobolev space. Compactly supported and local Sobolev spaces. Symbols of differential operators. Statement of Elliptic Regularity II.
Reading: §§12.1.3–12.2.2.
Pset 7, due Thu Nov 10: Exercises 12.1, 12.2, 12.3, 12.4, 12.5*, 12.6, 12.7, 12.8*, 12.9, 12.10.
Thu Oct 27 Fourier transform of compactly supported distributions. Fourier transform on L2. Sobolev spaces.
Reading: §§11.2.3–11.2.4,12.1.1–12.1.2.
Tue Oct 25 Fourier transform on Schwartz functions. Fourier Inversion Formula. Fourier transform on tempered distributions.
Reading: §§11.1–11.2.2.
Pset 6, due Thu Nov 3: Exercises 11.1, 11.2*, 11.3, 11.4, 11.5, 11.6, 11.7, 11.8, 11.9*, 11.10, 11.11*.
Thu Oct 20 A fundamental solution of the wave operator (in 1+3 dimensions).
Reading: §10.2.
Tue Oct 18 Pullbacks of distributions by submersions.
Reading: §10.1.
Pset 5, due Thu Oct 27: Exercises 9.1, 9.2, 9.3, 9.4, 9.5*, 10.1*, 10.2, 10.3, 10.4, 10.5, 10.6*, 10.7.
Thu Oct 13 Convolution and singular support. Fundamental solutions of constant coefficient differential operators. Examples: Laplace operator, Cauchy–Riemann operator, heat operator, wave operator (in 1+1 dimension). Elliptic Regularity I.
Reading: §§8.3, 9.1–9.2.
Thu Oct 6 Transpose of an operator and defining operators on distributions by duality. Convolution of two distributions.
Reading: §§7.3, 8.1–8.2.
Tue Oct 4 Tensor product of distributions. Operators between spaces of distributions and the Schwartz kernel theorem.
Reading: §§7.1–7.2.
Pset 4, due Thu Oct 20: Exercises 6.2, 7.1, 7.2*, 7.3, 7.4, 7.5, 7.6*, 8.1, 8.2, 8.3.
Thu Sep 29 Convolution of a distribution and a smooth function. Approximation of distributions by test functions.
Reading: §§6.1–6.2.
Tue Sep 27 Homogeneous distributions. Extending homogeneous distributions through the origin. Homogeneous distributions on the real line.
Reading: §§5.1–5.2.
Pset 3, due Thu Oct 6: Exercises 4.2, 4.3, 4.4, 4.5, 5.1, 5.2, 5.3, 5.4*, 6.1.
Thu Sep 22 More on support of distributions. Distributions with compact support as the dual of the space C. Fréchet metric on C and Banach–Steinhaus theorems for distributions. Distributions supported at a single point.
Reading: §§4.1–4.4.
Tue Sep 20 Differentiating distributions and multiplying them by smooth functions. Solving basic ODEs: u'=0 and xu=0. Support of a distribution.
Reading: §§3.1–3.2, 4.1.
Pset 2, due Thu Sep 29: Exercises 2.5*, 3.1, 3.2, 3.3, 3.4*, 3.5, 3.6, 4.1.
Thu Sep 15 Dirac delta function. Convergence of test functions and of distributions. Definition of a distribution via sequential continuity. Restrictions of distributions and the sheaf property.
Reading: §§2.1–2.3.
Tue Sep 13 The spaces Ck and C. Convolution. Approximation of rough functions by smooth functions. Partitions of unity. A function is determined by its integrals against test functions. Integration by parts. Definition of a distribution.
Reading: §§1.2.3–2.1.
Thu Sep 8 Prologue: motivation for distributions and elliptic regularity. Review: the spaces Lp, Riesz representation theorem for L2.
Reading: §§1.1–1.2.2.
Pset 1, due Thu Sep 22: Exercises 1.1, 1.2, 1.3, 2.1, 2.2, 2.3*, 2.4*.

Last updated: Dec 8, 2022