FMA350F

Temporary

Uncertainty Principles in Harmonic Analysis

7.5

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

2026-01-15

/Jonas Johansson

English

If sufficient demand

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.

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.

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.

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

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

Lectures

Seminars

Seminars given by participants

Failed, pass

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.

 

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.

FMA350F

2026-01-15

/Jonas Johansson