UBC Mathematics is pleased to announce that Professor Pablo Shmerkin, together with collaborator Tuomas Orponen (University of Jyväskylä), has published an article that has proven to be a key component in the solution of the long-standing Furstenberg set problem, which was recently solved by NYU Courant Institute of Mathematical Sciences Professor Hong Wang and Princeton University graduate student Kevin Ren. Professor Shmerkin’s results were recently published in the Journal of the American Mathematical Society (JAMS) and have already led to new developments in harmonic analysis and fractal geometry.

The Furstenberg set problem arose from the work of Abel Prize winner Hillel Furstenberg in the 1970s and was explicitly formulated by celebrated mathematical analyst Thomas Wolff in 1999. In geometric measure theory and harmonic analysis, this problem roughly asks: if a set of points contains small line segments in many different directions, how large must the entire set be?

Professor Shmerkin and his collaborators approached the Furstenberg set problem by refining multiscale decomposition—a technique that approaches a global problem by analyzing it locally across many scales. In their approach, the scales are selected more deliberately, guided by a combinatorial analysis of the object being studied. This method allowed the Furstenberg problem to be broken down into more manageable pieces.

Reflecting on his key role in solving the Furstenberg set problem, Professor Shmerkin shared, “Its resolution was the outcome of gradual progress across a decade and multiple collaborations in which we overcame the stumbling blocks one by one.”

For his work on the Furstenberg set conjecture, Professor Shmerkin, along with collaborators Tuomas Orponen (University of Jyväskylä), Joshua Zahl (Nankai University—on leave from UBC), and Hong Wang (IHES and NYU), has been awarded the 2026 Clay Research Award, which he will receive at the Clay Research Conference in September 2026.