Hong Wang, Member (2019–21) in the School of Mathematics, together with Joshua Zahl of the University of British Columbia, has announced a solution of the three-dimensional Kakeya conjecture. Their most recent paper, posted to arXiv in February, is the third in a series that relates to this breakthrough.
This long-standing conjecture is easily stated but has eluded mathematicians for decades: Take a thin laser beam that needs to shine in every possible direction in space. What is the smallest volume through which this beam must pass?
In two dimensions (like on a flat wall), the beam can point in all directions while passing through an area that is almost zero, but such a set cannot be too small, in that it must have maximal mathematical dimension—namely two.
But what about in our 3D world? The Kakeya conjecture proposed that in three dimensions, you also cannot be as efficient—there is a fundamental limit to how compactly you can arrange all possible directions in 3D space. This is precisely what Wang and Zahl have proved: the dimension of such a Kakeya set must be three.
Their work builds on that of other IAS scholars. Jean Bourgain, Professor (1994–2018) in the School of Mathematics, established important lower bounds for the size of Kakeya sets in multiple dimensions, proving that they could not be as small as known previously. His work also made fundamental connections between the Kakeya problem and other areas of mathematics. Zeev Dvir, frequent Member in the School, resolved an analogue of the Kakeya problem in an algebraic setting using the “polynomial method,” a powerful technique from algebraic geometry. Larry Guth, Member (2010–11), added a vital perspective related to this method in the setting of real three-dimensional space.
In addition to this, in the first of their papers, published in 2022, Wang and Zahl employed a technique pioneered by Terence Tao, Member (2005, 2023), and his colleague Nets Katz. In 2014, Tao and Katz showed that a specific subset of Kakeya sets must occupy a full three-dimensional size, exactly as the conjecture predicted. Wang and Zahl’s second and third papers extend this by dealing with all possible Kakeya sets.
Although Wang and Zahl have resolved a landmark open problem, further work remains. The conjecture about Kakeya sets has been made for all dimensions, and in higher dimensions, the geometry becomes more complicated.
Techniques developed for solving a problem in one area of mathematics often yield insights in another. Thus, the Kakeya problem is closely connected to important restriction conjectures about Fourier transforms, as first highlighted in the work of Princeton University’s Charles Fefferman. It is also related to problems in homogeneous dynamics, as demonstrated in recent works of Elon Lindenstrauss, Professor in the School of Mathematics, and his collaborators.
