Yes, presumably this:

https://terrytao.wordpress.com/2025/02/25/the-three-dimensio...

It doesn't help what we've designed a rather silly academic system where principal investigators are forced to spend a good deal of their time thinking where they'll be applying for their next grant. We also optimise the system for short term thinking rather than long games. There are some exceptions in research institutes but I think young people are the ones who have the clearest minds because of it.

Whenever I hear this claim about younger mathematicians I wonder if it still holds true (or really did historically). For example, Andrew Wiles proved Fermat’s Last Theorem in his 40s and there are numerous examples of productive older mathematicians. But also I think the claim skews towards big flashy problems rather than the work of building mathematical frameworks, finding structural insights and finding connections between disparate areas (which requires broad experience rather than just young intensity).

I think this hasn't been true in a long, long time. The most recent example of major contributions coming from someone in their 20s would be Evariste Galois around the time of the French Revolution.

Teens? No way, not really ever.

https://en.m.wikipedia.org/wiki/Shoshin

https://en.m.wikipedia.org/wiki/Einstellung_effect

Probably it is because the first solved problem seems like fun, but solving problems daily as a job quickly become boring?

The Fields Medal has a cutoff age of 40 years old.

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Anything original, for that matter.

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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.

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.

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.

A PhD is as much a stamp of endurance as it is a stamp of intelligence or accomplishment.

There is no deep theory here, bureaucracy doesn't think deep.

A PhD can be an opportunity to learn, or an opportunity to brag. I'm guessing the teen in the article is going to go for the first one.

Great question. I have a PhD. People forgot the purpose of a PhD. Hannah effectively achieved what many with a PhD fail to do, and that is contribute novel research. A PhD in the US (only place I can comment on) has lately been focused first and foremost on a) preparing for academia, which entails teaching and a lot of courses, and b) research for industry positions (many students in my cohort were from China or India and this was their segway into a job in the US). I agree a PhD should be purely focused on research and extending human knowledge. In practice, it is a business where students go to conferences to promote their PI's work, where Universities get cheap lecturers in the form of TAs, and where many mediocre students write incremental papers to secure an RnD position (change this by a little and see how it affects your results. This is your paper). I am very impressed by Hannah's work though and she embodies the selfless nature of research that is very much missing. I see too often people seeking to advance their own career and pick a PhD route of least resistance. While they are entitled to maximize profits, and oftentimes do not want to go to academia where solving the impossible is admired, we must remember discoveries often hinge on challenging problems and a selfless pursuit of the impossible. This is just my opinion based on what I saw in my cohort and at 30+ conferences

I remember at first year of university being presented with a bunch of “simple” problems early on, such as the Collatz conjecture.

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.

I give all of my hard problems to juniors.

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-...

Carl Gauss, Mozart, Blaise Pascal - his famous theorem was at 17, I believe

“Child prodigies” lists some more candidates https://en.wikipedia.org/wiki/List_of_child_prodigies

Galois Theory was a third year second semester course at my university when I studied there, which corresponds to 20-21yo in the UK system.

I would assume by having parents involved one way or another early on, with things like private, 1:1 education or the parents themselves being researchers

You also need the young person needs to also be extremely motivated and have what it takes. So both the contextual and individual means.

>This article is quite poorly written.

Her last name is misspelt in the very first paragraph as well.

Oh come on. This is in the Spanish newspaper El Pais. Context and audience matters. It’s simultaneously news about a math problem, an article about a young mathematician, and an article about things that happened at a math conference in Spain, which is where they presumably interviewed her.

> It should have been simple to add a line about the "other important results" and not violate show not tell.

Only a very tiny fraction of the publication's readers will have any idea what the heck the Mizohata-Takeuchi conjecture is. Naming the important results that Mizohata–Takeuchi being true would have validated would have been just technobable to nearly everyone reading the article.

I submitted under an approximation of the original title, and it was edited within seconds.

Yes, either proving a true conjecture or refuting a false one is "solving" it.

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.

Yes, found a counterexample to the conjecture.

Harmonic analysis?

Could you please stop posting unsubstantive comments to Hacker News? we're trying for something different here.