This lecture is about Fourier uniqueness and Fourier interpolation pairs, some explicit constructions of such pairs and their applications. Suppose that we have two subsets X and Y of the Euclidean space. Can we reconstruct a function f from its restriction to the set X and the restriction of its Fourier transform to the set Y? We are interested in the pairs (X,Y) such that the answer to the question above is affirmative. In this talk I will give an overview of recent progress on explicit constructions and existence results for Fourier interpolation pairs and corresponding interpolation formulas. Also I will try to convince you that explicit Fourier interpolation is a useful gadget in solving optimization problem and analysing differential equations.

Maryna Viazovska completed her Bachelor's degree at the Kyiv National Taras Shevchenko University and graduated with a Master's degree at the Technical University of Kaiserslautern. In 2013 she obtained her doctoral degree in Bonn followed by postdoctoral positions at the Institut des Hautes Etudes Scientifiques and the Humboldt-Universität zu Berlin. In 2017, Maryna was Distinguished Visitor at Princeton University. She joined EPFL as a tenure-track assistant professor in 2017 and was appointed full professor in 2018. In autumn 2020, she received the Latsis prize for her solution to a centuries-old problem about the optimal arrangement of spheres in a defined space.