The semester will focus on the intersections between modern harmonic analysis, number theory as well as arithmetic and geometric Ramsey theory, which are still evolving. In the past 20 years, there have been significant advances in these fields, resulting in the resolution of longstanding conjectures. The goal of this event will be to promote further interaction between these fields by highlighting methods of discrete, classical and higher order Fourier analysis, that drives these interactions.
Discrete methods in analysis have been extensively developed in various contexts over the past century. This line of inquiry began in the 1920s with the Hardy--Littlewood maximal inequality, and has seen profound extensions following the work of Bourgain in the late 1980s on pointwise polynomial ergodic theorems and on non-linear periodic PDE. Nowadays, higher-order Fourier analysis and discrete harmonic analysis have become independent disciplines that intertwine with each other, having profound impact on the development of modern ergodic theory, number theory and combinatorics.