On a simple sheet placed flat, a needle that turns on itself does not seem to upset anything. However, behind this elementary movement hides one of the most feared puzzles of modern mathematics. For more than a century, Kakeya's conjecture has been challenging researchers and intuitions, asking an apparently banal question but with dizzying ramifications. It is in space, beyond two dimensions, that this rotation becomes a matter of fractal geometry, of hidden dimensions and theoretical breakthroughs which affect artificial intelligence.
These mathematical objects, although devoid of area according to the measure of Lebesgue, are nonetheless rich in properties. When projected in three -dimensional spaces, the usual rules are no longer enough. This is where complexity explodes, and with it the interest of researchers. Because it is no longer the surface, but the very dimension of these sets which becomes the center of all attentions.
Why the conjecture of Kakeya has fascinated mathematicians for a century
What is called today the conjecture of Kakeya claims that a set of Kakeya, in a dimensional space, has a dimension of Hausdorff and Minkowski equal to n. An assertion that seems natural, but whose demonstration has proven to be extraordinarily difficult. However, it was verified for cases to one and two dimensions from the 20th century. But from the third dimension, the evidence failed one after the other.
It is this impasse that Hong Wang, professor at New York University, and Joshua Zahl, professor at the University of British Columbia. Their work, published on the scientific server Arxiv in February 2025, is based on an innovative approach combining multi-scale analysis, geometry of tubes, and so-called non-concentration conditions. They demonstrate that even in three dimensions, a set of Kakeya retains a complete structure, with a dimension of 3 according to the two mathematical definitions.
This result is not just a academic victory. It also validates a long -standing intuition in some researchers such as Larry Guth or Terence Tao, for whom the conjecture seemed to be unassailable but fundamentally true. The New Scientist site greets such a powerful demonstration that it could become the key to other puzzles so far deemed insoluble.
From concrete applications to cryptography and artificial intelligence
Because this needle that pivots in space entails with it a whole section of modern mathematics. By proving the conjecture of Kakeya in R³, Wang and Zahl reinforced the foundations of harmonic analysis, a crucial area to understand the propagation of waves, differential equations and even certain structures of artificial intelligence. In Quanta Magazine, Jonathan Hickman recalls that Kakeya sets are linked to three key conjectures in analysis: restriction, Bochner-diez and local smoothing. These problems, formerly considered almost independent, are actually based on the same geometric base.
The consequences also extend to the field of cryptography. The precise study of the sets of tubes, at the heart of this demonstration, directly influences the way in which the data amounts or circulates in the digital space. As New York University explains, these advances could improve certain cybersecurity tools. They are based on long neglected geometric properties, but today carrying real potential.
Finally, this work demonstrates the power of the induction method on the scales, described in detail in their study. A technique that could be reused to face other famous conjectures, including that of Riemann according to Iflscience. If the conjecture of Kakeya was only one obstacle among others, its erasure now leaves room for a new generation of ambitious and transversal evidence.
