Joshua Zahl

A Clay Research Award is made to Tuomas Orponen (Helsinki), Pablo Shmerkin (UBC), Hong Wang (IHES and NYU), and Joshua Zahl (Nankai) in recognition of their remarkable work on geometric problems in harmonic analysis, leading to the proof of the Furstenberg set conjecture in the plane and the Kakeya conjecture in three dimensions.

The Furstenberg set conjecture is a fundamental problem about the intersection patterns of thin tubes in the plane.  It connects to many areas of mathematics.  It answers basic questions in projection theory that were raised by Kaufman in the 1960s.  Furstenberg raised the problem in the late 60s because of connections to ergodic theory.  It can also be viewed as a continuum version of the Szemeredi-Trotter theorem in combinatorics.  And Wolff studied it in the 90s because of connections to harmonic analysis. In addition to the Award winners,  Kevin Ren also made a substantial contribution to its resolution.

The Kakeya set conjecture is a fundamental problem about the intersection patterns of thin tubes in space.  Fefferman’s work on the ball multiplier conjecture showed that the Kakeya problem is a key roadblock for a range of open problems in Fourier analysis, including Stein’s restriction problem and the local smoothing problem for the wave equation.

These results build on a new set of tools for multiscale analysis developed by these four mathematicians (and some others) over many papers.  Older work in the field often described the geometry of a set in Euclidean space using just one number, such as the Hausdorff dimension of the set.  Instead, the new work considers detailed information about the spacing of the set at each scale.  Different spacing scenarios are exploited in different ways.

Photo by Paul Joseph