Yes, presumably this:

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

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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.

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

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.

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

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

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 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-...

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.