Mathematicians have resolved a decades-old geometric problem, the Kakeya conjecture in 3D, which studies the shape left behind by a needle moving in multiple directions.
The Kakeya conjecture was inspired by a problem asked in 1917 by Japanese mathematician Sōichi Kakeya: What is the region of smallest possible area in which it is possible to rotate a needle 180 degrees in the plane? Such regions are called Kakeya needle sets.
“It stands as one of the top mathematical achievements of the 21st century.”
Hong Wang, an associate professor at New York University’s Courant Institute of Mathematical Sciences, and Joshua Zahl, an associate professor in the University of British Columbia’s mathematics department, in an article recently posted to the preprint server arXiv, which hosts research before it is peer-reviewed and published in a journal, have shown that Kakeya sets, which are closely related to Kakeya needle sets, cannot be “too small.”
Namely, while it is possible for these sets to have zero three-dimensional volume, they must nonetheless be three-dimensional.
“There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture!” writes UCLA mathematics professor Terence Tao, who won the 2006 Fields Medal, which is awarded every four years to a mathematician under the age of 40.
“It stands as one of the top mathematical achievements of the 21st century,” says Eyal Lubetzky, the chair of the mathematics department at the Courant Institute.
“This is a wonderful piece of mathematics,” adds Courant Institute Professor Guido De Philippis.
“The latest work follows years of progress that has enhanced our understanding of a complicated geometry and brings it to a new level. I am expecting that their ideas will lead to a series of exciting breakthroughs in the coming years!”
“This is a problem that many of the world’s greatest mathematicians have worked on, and for good reason—in addition to having the appeal of being relatively simple to state yet extremely deep, it is connected to many other major problems in harmonic analysis and geometric measure theory,” says Pablo Shmerkin, a professor of mathematics at UBC.
“While building on recent advances in the area, this resolution combines many new insights together with remarkable technical mastery. For example, the authors were able to find a statement about tube intersections that is both more general than the Kakeya conjecture and easier to tackle with a powerful approach known as induction on scales.”
Proving the Kakeya conjecture requires a fine understanding of the structure of the interaction of tubes in Euclidean—three-dimensional—space.
“This result is not only a major breakthrough in geometric measure theory, but it also opens up a series of exciting developments in harmonic analysis, number theory, and applications in computer science and cryptography,” adds De Philippis.
“Indeed in several problems in these fields, relevant information can be decomposed into wave packets—regions of space where electromagnetic or other types of waves are located—which are largely concentrated on ‘tiny tubes.’ Understanding the intersection of these tubes is fundamental in understanding how these packets of information interact one with the other.”
Source: NYU
