Uncertainty Principles in Harmonic Analysis
Osäkerhetsprinciper i harmonisk analys

FMA350F, 7.5 credits

Valid from: Spring 2026
Decided by: /Jonas Johansson
Date of establishment: 2026-01-15

General Information

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

Aim

The overall goal of the course is to introduce participants to uncertainty principles in harmonic analysis and show how these issues relate to various problems in mathematical analysis. The course thus provides a comprehensive overview of how various threshold phenomena in harmonic and complex analysis are related to other mathematical disciplines.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to analyze mathematical questions based on methods developed in connection with uncertainty principles in harmonic analysis,
• be able to give examples of important applications of the uncertainty principles, as well as the methods that form the basis for these,
• be able to give a detailed account of the theory behind the methods introduced in the course,
• be able to give an account of research problems within the subject and relate these to uncertainty principles.

Competences and Skills

For a passing grade the doctoral student must

• integrate knowledge from different areas of mathematics to analyze the relevant issues that arise when studying uncertainty principles
• independently identify, formulate, and communicate these mathematical ideas
• plan and execute qualified presentations of the course content within a given time frame
• give oral presentations on advanced mathematical concepts

Judgement and Approach

For a passing grade the doctoral student must

• evaluate different methods from uncertainty principles in harmonic analysis and what roles they play in different research areas in mathematics and physics
• give constructive criticism on other students' ability to communicate mathematical ideas in relation to uncertainty principles

Course Contents

The course treats a collection of threshold phenomena in harmonic analysis, such as completeness problems and uniqueness problems. This includes topics such as Wiener's theorem, Heisenberg's uncertainty principle, and Ivashev-Musatov's theorem. The course provides a comprehensive summary of the theory of Nevannlinna theory from different perspectives on uncertainty principles, with applications to F. and M. Riesz's theorems, Szegö's theorem, and Khrushchev's theorem. Much of the course content will touch on elements from other mathematical disciplines, such as probability theory, operator theory, potential theory, and fractal geometry.

Course Literature

The teacher's own compendium will be used, it is largely based on the book The Uncertainty principle in Harmonic Analysis by V. Havin and B. Jöricke.

Instruction Details

Types of instruction: Lectures, seminars

Examination Details

Examination format: Seminars given by participants
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: Fourier analysis, Analytic functions, Integration theory, Linear functional analysis, or comparable prior knowledge is required. Hardy space and Harmonic analysis are recommended, but are not required.

Course Occasion Information

Contact and Other Information

Course coordinators: