‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture | Quanta Magazine
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.
Hasnain says:
“The conjecture’s resolution is a seismic shift for the field of harmonic analysis, which studies the details of the Fourier transform.
A tower of three monumental conjectures in harmonic analysis rests atop the Kakeya conjecture. Each story in the tower needs to be sturdy for the stories above it to stand a chance themselves. If the Kakeya conjecture had been proved false — if Wang and Zahl had found a counterexample — the entire tower would have come tumbling down.
But now that they’ve proved it, mathematicians might be able to work their way up the tower, using Kakeya to build up proofs of these successively more ambitious conjectures. “All these problems that [mathematicians] dreamed about someday solving, they all look approachable now,” Guth said.”