Georgia Institute of Technology College of Sciences

Harmonic Analysis witnessed a number of breakthroughs in recent years. The most famous one is perhaps the Kakeya conjecture in three dimensions, recently solved by Wang and Zahl. Very roughly speaking, the major conjectures in the field are either of oscillatory or non-oscillatory nature. A major modern challenge is to build bridges between these universes by using tools of the latter kind to solve problems of the former. The goal of this talk is to take a walk along the constellation of (yet) open problems in Harmonic Analysis and on the bridges that connect them. One such bridge, Stein’s conjecture for weighted L^2 Fourier extension estimates, was recently shown to have a few 'cracks' and raised concerns that it may be broken, and we will see these cracks by putting together the Schrödinger equation, counts of rectangles and isosceles triangles, and elementary number theory.