This online mini-course by Mikhail Sodin (Tel Aviv University) is intended for advanced undergraduate and graduate students from Ukraine. The prerequisites are: basics of the Fourier series and Fourier transform, the Lebesgue integration, L^2 spaces, complex analysis and basic operator theory.

Lectures will be given in English.

We plan these lectures as an invitation to a classical area of analysis, that continues to be an active area of research with new discoveries emerging. We hope to discuss the following topics:

• Heisenberg uncertainty principle: This principle states that a function and its Fourier transform cannot be simultaneously well-localized.

• Hardy theorem: This theorem states that if a function and its Fourier transform have the standard Gaussian decay at infinity, then it is a Gaussian function.

• Benedicks theorem: This theorem states that if a function and its Fourier transform are supported by a set of finite Lebesgue measure, then it is the zero function.

• Radchenko-Viazovska-type uniqueness pairs: These are discrete sets A and B such that if a sufficiently smooth and fast decaying function vanishes on A and its Fourier transform vanishes on B, then it is the zero function.

Mikhail Sodin has been a professor of mathematics at Tel Aviv University since 1996. He graduated from V. N. Karazin Kharkiv National University in 1979 and obtained his doctoral degree at the Institute of Mathematics of Armenian Academy of Sciences in 1985. Mikhail worked at the Kharkiv Institute of Radio Electronics (1979-1989) and Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine (1989-1996). Mikhail Sodin is one of leading analysts in the world. Among his many mathematical accomplishments, together with Nazarov he introduced new fundamental tools to study random spherical harmonics.