Also, Terence Tao hinted at some further advances some time ago [2], does anyone know more about that?
https://terrytao.wordpress.com/2025/02/25/the-three-dimensio...
Teens? No way, not really ever.
This article is quite poorly written. Case in point above. If the conjecture was believed to be true, refuting it would be news in itself, deserve more than half a sentence, and have nothing to do with the age of the refuter. It should have been simple to add a line about the "other important results" and not violate show not tell. AlsO I fail to see the relevance of mentioning the Spanish academy? The researcher is from Bahamas/USA, it's just the writer is from Spain?
Her last name is misspelt in the very first paragraph as well.
> Cairo solved the so-called Mizohata-Takeuchi conjecture, a problem first proposed in the 1980s that had kept the harmonic analysis community had been working on for decades. The conjecture was widely believed to be true — if so, it would have automatically validated several other important results in the field — but the community greeted the new development with both enthusiasm and surprise: the author was a 17-year-old who hadn’t yet finished high school.
The writer introduced and resolved the potential disappointment much more elegantly and in far fewer words than I can manage here by paraphrasing. I admire that and feel it's indicative of good writing, albeit spoiled by an earlier typo.
Euler was 41 when he discovered his famous identity, the kind of thing people learn in school.
Even Newton was 21 when he invented calculus, the sort of stuff that you might find late teens learning.
Galois by a couple of years? He died at 20, and I suppose they teach that stuff sometime mid uni?
There is a lesson there: always give people an opportunity to excel, if you can.
I remember wanting to spend time trying to explore what a solution might look like, because such simply formulated problems must have equally simple solutions.
Maturing and getting a better understanding of my intellectual capacity, I have opted to solve practical problems with a much bigger chance of success and absolutely no groundbreaking qualities.
But I liked being taken serious from the start, and I think it’s important to try and solve hard problems before you grow stuck in the real world.
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.
https://www.nytimes.com/interactive/2025/06/30/science/math-...
The Poynting-Vector (https://en.wikipedia.org/wiki/Poynting_vector) indicates the direction of the energy flow of a magnetic field, but it’s named after a physicist called „Poynting“, not because it is „pointing“ somewhere. I thought the „y“ in poynting was a typo when I first read it.
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?
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.
I hope she's found a program that will support her while realizing she's smarter than whomever is setting the rules, rather than something stifling.
At 17 she’s so young that a uni hiring her would need to think about child labor laws
Many institutions would actually jump at the chance. That's way better than a 35 or 37 year old burnt out from just finishing their PhD and getting onto the tenure track suffer-fest. Think of how many years of productive research she has in her. It used to be way more common until academia became so professionalized and bureaucratic.
> “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.
PhDs "by publication" refers to not having to submit work additional that already published to the examining committee.
Do you have a PhD from a theory department? I do. You're wrong.
Nobody gets a PhD by publication without publications. It's literally axiomatic.
> It's literally axiomatic.
You've made up some axiomatic definition of "by publication" that does not bear any resemblance to the actual definition. Consider that it's possible to
1. Submit a preprint to arxiv and have it count
2. Submit a preprint to a journal and defend before it accepted (or rejected)
3. Not submit anything anywhere and have the PhD itself count (almost all PhDs get an ORCID)
This is a process where you can write your "dissertation" by putting an intro and a conclusion on ~3 papers you have already published and get a PhD that way. You enroll in the school for ~3 months, write the missing parts, and that's it. This is a flexible path to a PhD for industry researchers or other people who have a lot of expertise and have pushed the boundaries of a field but did not do a formal PhD program.
I have never heard of anyone doing this with ArXiV preprints or any school accepting this path if they are not referreed papers. I would love to see an actual counterexample if you have one.
There are degree mills that do what you describe.
There is also the format in countries such as Germany or the Netherlands where one typically "bundles" one's publications into a thesis. However, the work is typically done in the context of supervised doctoral programmes and no less rigorous than that done under different PhD studies formats.
https://www.cambridgestudents.cam.ac.uk/exams/students/postg...
Like I said above, this is not usually offered unless you are clearly doing work of sufficient quality. 3 ArXiV preprints can get you a PhD just fine, but it won't cut it if you wrote them when nobody is watching.
On average someone that does have a PhD will have a wider set of skills, like understanding of the complexities of the field, resistance to frustration, capability to do research and ability to communicate.
"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."
[1] https://web.archive.org/web/20250706185220/https://news.ycom...
Showing that there exists an x such that the statement is false disproves the conjecture.
She found a counterexample.
Article could at least spell her name correctly.