[arutar]

This is a very phenomenal result and everyone in the field is excited about this! Josh and Hong both gave talks about this at a conference 2 weeks ago in Berkeley, and the videos are online: [1] [2]. Josh's talk (at least from my perspective of someone who is adjacent to this field) is quite approachable, whereas Hong talks more about the induction scheme on Guth's grains decomposition which is quite a bit more technical.

I visited Josh at UBC last year around this time and I recall asking him if he thought Kakeya in dimension 3 would be solved soon. I remember that he believed it would be (though perhaps he was worried by someone other than Hong and himself). In the end they were able to complete the proof themselves.

Hong is probably quite a serious candidate for the fields medal because of this. She has already made impressive progress on problems in harmonic analysis and geometric measure theory and she is one of few people has a firm foothold in both fields at the same time.

The quanta article talks about a 'tower of conjectures' in harmonic analysis; at the top of the tower is the so-called "local smoothing conjecture" (a conjecture about how much waves, such as 'idealized' sound waves, can amplify from some initial configuration when averaged over time). A Kakeya set is a certain type of geometric obstruction to local smoothing; resolving the full conjecture also requires handling so-called 'oscillatory' obstructions. In dimension 2 + 1 (2 spatial and 1 time dimension) the local smoothing was only recently resolved (also by Hong and co-authors [3]); even though the corresponding result for Kakeya sets in dimension 2 has been known for over 40 years.

[1] https://player.vimeo.com/video/1062254156

[2] https://player.vimeo.com/video/1063428579

[3] https://annals.math.princeton.edu/2020/192-2/p06