Hannah Cairo has solved the Mizohata-Takeuchi conjecture
27 comments
·July 6, 2025marvinborner
mellosouls
Yes, presumably this:
https://terrytao.wordpress.com/2025/02/25/the-three-dimensio...
ledauphin
here's a dumb question:
she's starting her Ph.D. this fall - hasn't she already achieved it? What is the theory behind expecting someone who has solved a decades-old problem to do some "second" thing to prove that they have extended the bounds of human knowledge?
EvgeniyZh
Ph.D. is training in how to do research. Solving one, even very hard problem not necessarily means that you don't need such training. It's especially tricky with counterexamples which sometimes question of raw talent and luck rather than skill.
The next step for someone who has PhD and want to stay in academia is postdoc. After solving one problem, you would not necessarily have what's needed to get a good postdoc, such as clear research agenda or proof of ability to publish consistently.
parpfish
But what does somebody do with a PhD at age 17? I can’t imagine hiring them as a prof when they’re so young. It’s not a bad idea to just take a couple years to continue your already productive collaboration while getting mentored on the non-math parts of being a mathematician.
ics
IIRC Erik Demaine (https://en.wikipedia.org/wiki/Erik_Demaine) started teaching at 20 and had his PhD. I can't remember if I first saw his name because of the MacArthur Grant or one of those science documentaries but one of his pages was on the frontpage here a week or two ago and it seems like he's been thriving.
nextos
A PhD in the US requires a lot of coursework, aside from research. Perhaps, she is interested in that. Otherwise, some universities, especially in EU, offer PhDs by publication. She could simply wrap up her counter-example publication (https://arxiv.org/pdf/2502.06137) as a thesis and possibly graduate. Sometimes, you can even do this without a supervisor.
xg15
Sounds as if she even has a potential supervisor:
> “It took me a while to convince Ruixiang Zhang [the professor of the course where the problem had been posed] that my proposal was actually correct,” Cairo says
> At the University of Maryland, she will continue working under the supervision of Zhang. “He helped me so much, and I’m really grateful. Beyond his class, which I loved, he spent countless hours tutoring me,” she recalls.
null
raincom
Great achievement. Now Princeton Math department will ask her to join their school for Ph.D.
Keyframe
Original title is more informative than the edited one here.
leephillips
I submitted under an approximation of the original title, and it was edited within seconds.
miles
There is too much "helpful" title modification of late. The original title itself fits within HN limits:
"A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago"
The site's guidelines are clear[1] but increasingly ignored by some moderators:
"...please use the original title, unless it is misleading or linkbait; don't editorialize."
scythe
Paper here:
https://arxiv.org/abs/2502.06137
I had the opportunity to take a harmonic analysis course in grad school. I passed it up. It was only tangentially related to my research at the time.
munchler
I had never heard of the X-Ray Transform until I happened to read about it in the New York Times today, and then here it is again.
https://www.nytimes.com/interactive/2025/06/30/science/math-...
dekhn
This sort of transform (what I think many people call inverse problems) is quite common in reconstruction problems- that is, where you pass light or other EM through an object, the light scatters, and hits a detector. Typically you want to find the minimum error reconstruction. See more here: https://en.wikipedia.org/wiki/Radon_transform
paulpauper
Trying to do anything original and novel in math is extremely hard at any age. to do it at 17 is insanely talented. congrats
pillefitz
Anything original, for that matter.
pinoy420
[dead]
kemitchell
Refuted?
zahlman
Yes, either proving a true conjecture or refuting a false one is "solving" it.
qsort
The Mizohata-Takeuchi conjecture is a statement in the form "For all <x> (a bunch of math)".
Showing that there exists an x such that the statement is false disproves the conjecture.
She found a counterexample.
mgiampapa
She found more than one way of disproving it in the process.
gilleain
Yes, found a counterexample to the conjecture.
tomjen3
>One day, he proposed proving a special, much simpler case of the conjecture as a homework assignment. As an optional part, he included the original conjecture
There is a lesson there: always give people an opportunity to excel, if you can.
old_man_cato
[flagged]
There's a video by Hannah Cairo that explains the conjecture and her results [1]
Also, Terence Tao hinted at some further advances some time ago [2], does anyone know more about that?
[1]: https://www.youtube.com/watch?v=3ZeH_8sTyKA
[2]: https://mathstodon.xyz/@tao/114003793236630744