Posted on 03/30/2025 9:56:54 AM PDT by Red Badger
Hong Wang, an associate professor at NYU’s Courant Institute of Mathematical Sciences, presenting her work on the Kakeya conjecture on March 10, 2025. Credit: David Song/NYU.
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Professors from NYU and the University of British Columbia have resolved the Kakeya set conjecture in three dimensions.
Mathematicians from New York University and the University of British Columbia have resolved a long-standing geometric problem known as the Kakeya conjecture in three dimensions. This conjecture explores the minimal space required for a needle, or line segment, to point in every direction within a given space.
The idea originates from a 1917 question posed by Japanese mathematician Sōichi Kakeya: What is the smallest area in the plane where a needle can be rotated 180 degrees? The regions that allow such movement are called Kakeya needle sets.
Hong Wang, an associate professor at NYU’s Courant Institute of Mathematical Sciences, and Joshua Zahl, an associate professor in UBC’s Department of Mathematics, 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!” wrote 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.
Recognition from the Mathematical Community
“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.”
Broader Impacts Across Fields
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.”
Reference:
“Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions”
by Hong Wang and Joshua Zahl, 24 February 2025, arXiv.
DOI: 10.48550/arXiv.2502.17655
Stop the presses! I just did some quick calculations. There’s a serious error in the above. Partial results do NOT exist in higher dimensions.
Okay, I’ll admit it. I didn’t catch the error. My cat did.
Angels is apt. Most of mathematics is abstract, eternal, and really a branch of God, but has little to no practical use in the mortal world. It seems most articles of these discoveries these days try real hard to tie them to either quantum computing or global warming, but that is political funding BS. They should tie them to God instead, and give thanks for enough brains to discover a bit more of Him.
angels, or perhaps angles?
I think it’s time to make these mathematicians to walk the planck.
On the contrary, math is very useful in that it mimics nature.
So much so that science and technology co-opted it to make a myriad of shiny playthings.
Or it might turn out to be a useful tool to deal with a problem in another field - as sometimes happens.
I had that figured out to within a half inch.,,,,
You're obviously not a PHd mathematician who has nothing else to discuss with his peers on Friday nights during happy hour.....
Math is a wonderful thing it help many California drives pass the test.
3R = 1L
3 right turns equal on left turn
Math is a wonderful thing it help many California drives pass the test.
3R = 1L
3 right turns equal one left turn
fixed
Has anybody looked at California deficits lately. Let’s learn second grade arithmetic first
The proof was in the pudding.
I got the mathematician in my family a Kakeya needle set for Christmas last year.
Sometimes. Math can be used as a tool to create models of the physical world. However, 1) all models are wrong at some level, by definition, and 2) most models are failures, like the 80 models used to come up with the runaway man-made global warming scam, possibly the most expensive scam in history.
Math is interesting for being one of the few things that contains perfect truths. But nature is much more fuzzy, unknown, and for us mortals, unknowable.
A PhD in physics once told me that Pie r Round ...
The mystery of the universe is solved !!!
“You’re obviously not a PHd mathematician”
And obviously neither are you!
The correct abbreviation is as follows: “Ph.D.”
Tsk - tsk - tsk.
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