2026 Clay Research Awards
Date: 14 April 2026
2026 Clay Research Awards are made to Thomas Orponen, Pablo Shmerkin, Hong Wang, and Joshua Zahl; to Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank; and to Yu Deng and Zaher Hani.
Orponen, Shmerkin, Wang, and 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.
Burklund, Hahn, Levy, and Schlank
A Clay Research Award is made to Robert Burklund (Copenhagen), Jeremy Hahn (MIT), Ishan Levy (IAS and CMI), and Tomer Schlank (Chicago) in recognition of their remarkable construction of counterexamples to Ravenel’s “Telescope Conjecture.”
The Telescope Conjecture was the last open conjecture from Ravenel’s visionary paper “Localization with respect to certain periodic homology theories.” That paper, and the body of work it inspired, form the bedrock of chromatic homotopy theory. In one version, the telescope conjecture postulates an upper bound on the growth rate of the chromatic layers of the stable homotopy groups of spheres. The work of Burklund, Hahn, Levy and Schlank is the crest of a revolutionary new wave in K-theoretic techniques, to which they have each, independently, contributed. Their counterexamples imply that the p-rank of the stable homotopy groups of spheres grows faster than expected, and contains a proliferation of elements that are unaccountable by any prior understanding of the subject. This is a milestone achievement.
Deng and Hani
A Clay Research Award is made to Yu Deng (Chicago) and Zaher Hani (Michigan) in recognition of their remarkable derivation of the Boltzmann equation for long times, starting from a system of hard spheres.
The question of deriving macroscopic laws rigorously from microscopic models goes back at least to Hilbert’s 6th problem stated in 1900, and remains a deep and essentially unsolved problem in mathematical physics. Yu Deng and Zaher Hani, together with their co-author Xiao Ma, have solved part of the problem by deriving the Boltzmann equation describing intermediate, mesoscopic scales, starting from a microscopic system of hard spheres, for large times which may even diverge with the number of particles, as long as the regular solution of the equation exists.
The result involves an exceptional mastery in combinatorics and in designing algorithms in extremely intricate models, and is a breakthrough in the field, 50 years after Lanford’s seminal result for short times, and more than 150 years after Boltzmann’s long-debated theory.
