Course Syllabus for

Pseudodifferential Operators

FMA310F, 7.5 credits

Valid from: Autumn 2021
Decided by: Professor Thomas Johansson
Date of establishment: 2021-04-15

General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English


Throughout mathematics and physics, linear elliptic partial differential equations make their appearance. A natural class of operators arising in this context is that of pseudodifferential operators, which after the fact can be described in terms of the class of operators of the form A=f(D) for reasonable functions f and some elliptic partial differential operator D. In particular, the class of pseudodifferential operators contain parametrices and solution operators to elliptic partial differential equations. The purpose of the course is to introduce the details of pseudodifferential calculus, sometimes reducing deep questions about partial differential equations to first year calculus, and to study its consequence in the global analysis of elliptic partial differential equations.


Knowledge and Understanding

For a passing grade the doctoral student must

Competences and Skills

For a passing grade the doctoral student must

Course Contents

In this course we will study pseudodifferential operators with a view towards global results. Since the 1960s pseudodifferential operators have been used for the study of elliptic differential operators. The highlights from this include the Weyl law describing their spectral behaviour and Atiyah-Singer’s index theorem computing their index. After working through the basic methods of algebraic and analytic flavour, we study their operator theoretic consequences. We shall also consider applications in the study of boundary value problems and Hörmander’s proof of the Weyl law, asymptotically describing the eigenvalues of elliptic pseudodifferential operators.

Course Literature

Chapters XVIII and XX in Shubin's book and Chapters XVIII and XX in Hörmander.

Instruction Details

Type of instruction: Lectures

Examination Details

Examination formats: Oral exam, written assignments
Grading scale: Failed, pass

Admission Details

Assumed prior knowledge: Basic course on partial differential equations , e.g. FMAN55 Kontinuerliga system. Fourier analysis, basic functional analysis, distribution theory and differential geometry.

Course Occasion Information

Contact and Other Information

Course coordinators:

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