I will try to incorporate Hannah's notes style into my own stuff; I am using a similar lecturing device with the tablet, but her presentation is soooo far beyond mine. So beautiful!
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
Not really related but similar situation and I found it funny when I realized it:
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.
While extremely talented, I am not surprised to find this coming from a teen. Major mathematical discoveries often have come from those in their mid 20’s with the greater discoveries being skewed towards the younger 20s and teens. I think this because pure mathematics is just so creative.
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.
I wonder how many years on average people can dedicate themselves to think deep about research problems, before they must stop thinking and start managing, or quit the race.
You finish your degree, and start your PhD. The first year, you are busy learning techniques and getting caught up with the relevant literature.
You are far too concentrated on learning new things to get any thinking done.
In your second year of your PhD, you are getting better. You can do most things without thinking about them. This frees up your brain to think about other things.
However, your grasp of the wider literature is still lacking, so you use that brainspace to optimise your current experiments (as you should).
In your third year of your PhD, you are starting to write things up: either your thesis, or your first (big) paper. You read a lot more, you know a lot more.
The deep thinking can commence.
Your first postdoc is probably your most productive time: you know what you are doing; you know the state of the literature and which parts are reliable and which aren't; you have a clear idea of what problems need solving.
You are starting to write your first grant applications, but you only need one for yourself and not several to cover the needs of a full lab. You don't have any kids at home.
This is a good time to solve some big problems. It lasts about 2-3 years.
At the start of your second postdoc, you panic. The big problem was harder than you thought and you don't have enough high-impact papers to be competitive in job applications for a principal investigator (PI) role.
You start churning out low-value fillers and collaborating with everyone and their hamster to get your name on as many papers as possible.
The rest of the time is taken up by applying for grants and PI positions. You don't even make it to the interview stage. You start pondering about life outside of academia.
You could replace “research” with nearly any term not undergirded by a direct profit motive (eg civics, politics, education, community health, urban planning). Or maybe it’s just (and you may be saying this) that one can only dedicate themselves up until a clear profit opportunity appears.
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).
> there are numerous examples of productive older mathematicians
Curious about the extreme cases. Did any centenarians ever managed to come with an outstanding original math result? If it didn't happen before, I hope to see it happening in the next decades, given current demographic trends.
I was told that a book published in honor of Oscar Zariski's 80th birthday included a paper by Oscar Zariski, either proving or at least making progress on a longstanding conjecture by Oscar Zariski.
I was in the relevant department at the time (Harvard math), but I wasn't much of an algebraic geometer, so I took that at face value without probing for details.
That seems like a pretty weak argument. I don’t think he was just filling in easy details for the next decade plus. Also, even 33 is quite a bit older than the commonly claimed early 20s needed for great mathematical work.
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.
Peter Scholze received his Fields Medal at 30 for work he carried out through his 20’s. However he is not the youngest recipient: J.P. Serre received his Fields medal at 27.
Before we get too excited about the Fields Medal as an indicator for age of great mathematical achievements, let's remember that it's only awarded to people under 40 for work done earlier, potentially many years earlier.
My understanding is that that’s more about encouraging younger mathematicians rather than an expectation that older ones won’t produce anything worthy.
>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.
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 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.
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?
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.
Modern PhDs are not designed for people that are smart like this. She's a math savant that obviously has a unique and demonstrably effective way of looking at things, why destroy that with "training how to do research".
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.
This is not true at all. Maths PhDs are excellent for this kind of person, and foster and grow such abilities. No matter how smart you are, training on how to do research is going to improve your capabilities and success.
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.
> I can’t imagine hiring them as a prof when they’re so young
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.
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.
Noam Elkies too, the youngest ever tenured prof at Harvard. Another parallel he had a pretty famous contradiction proof, of Euler's conjecture. But he didn't find that until age 22, so seems like this girl has a good head start!
I was one of several math grad students who started at Harvard at age 16 or 17 aroud the same time. Ofer Gabber and Ran Donagi went on to conventional academic math careers. I took a less straightforward career path.
But I was offered an assistant professorship at the Kellogg School of Business at age 21, and have often wondered whether I should perhaps have taken that, or else the research position I was offered at RAND.
What does someone do with a PhD at age 35? Go into industry? Continue as a postdoc? Open a juice bar? It doesn't matter what she's going to "do" with it. It's accreditation of a certain degree of academic achievement, which she has achieved. Arguing that she doesn't deserve it or needs to earn it the "normal" way is stupid.
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.
> “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.
They must be peer-reviewed journal papers and I believe they tend to prefer if at least one is well-cited or significant, especially if you have only three papers. It is generally harder to get a PhD by publication than to get a PhD the normal way.
Nobody gets a PhD by publication with 0 publications. This is usually a backdoor for people who have done a lot of work in a field, certainly far more than a PhD thesis, and have just never gotten the credential.
It's amazing how many people on hn are experts on things they do not have the qualifications to be expert on.
> 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)
Are you aware that "PhD by publication" is a real thing that is a separate path than a normal PhD? It is relatively common for schools in some European countries to offer these, but not that common outside Europe.
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.
>You enroll in the school for ~3 months, write the missing parts, and that's it.
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.
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.
That's only available to those with an undergraduate degree from Cambridge who, subsequent to their undergraduate degree, publish work worthy of a Ph.D. and pass a viva.
I'm not sure how common a route ot Ph.D. it is (I never heard of it before), but it sounds like an anachronistic extension of the MA's Oxbridge graduates feel entitled to.
Correct. It is generally an exceptionally rare path to a PhD, but the door is open. I believe I saw the statement that this particular program has granted ~100 degrees over its life compared to ~15000 PhDs from Cambridge.
I agree, you seem to be claiming to be an expert on something.
"PhD by publication" is a specific thing, it's in italics in the previous post.
My Universities offer a "PhD by publication". You basically staple together a bunch of your publications, and write a brief intro. It saves you writing a full PhD document. But, the standard on those publications is quite high -- you certainly wouldn't get one from preprints to arxiv at any University I've ever worked at.
Of course, you can get a PhD with no publications, just write a good PhD. Lots of students do taht.
Did you know that there are universities outside of Europe? And that in those universities, "3 publications plus intro and conclusion for a PhD" is also called (usually) PhD by publication (sometimes it is called a kitchen-sink PhD). And my point was that that rule of thumb does not apply to theory students.
Note the italics please.
So I'm sorry you're right I'm not in expert in Europe's monopoly on the phrase "PhD by publication" but who would want to be an expert in trivial bs like that (answer: apparently you and the other guy!)
Maybe in a few days come back and re-read this thread. Maybe you are having a bad day, I don't know.
You are the one who started talking about people not being experts in having a PhD by publication. No-one else (including me) has bought up that they are an expert on that topic.
A PhD by publication sounds very similar to a higher doctorate (DSc, DLitt, etc). Substantive (as opposed to honoris causa) higher doctorates are awarded based on publication record only. To be eligible for a substantive higher doctorate, you generally are expected to have a PhD first - but it might not be an absolute requirement. You’d generally expect a bunch of papers, but in principle a single publication (if sufficiently groundbreaking) could be enough. While this is very impressive for a 17 year old (I wish I could have done that at 17, or at any other age for that matter), it probably isn’t significant enough for a higher doctorate all by itself. If she’d proved P=NP, different story. (Who knows, maybe she shall-well, probably not, but I’d be very happy to be proven wrong about that.)
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 do not think there is "one" purpose for a PhD, and not everyone gets the same thing from it - it depends on what are a person's strong and weak points. I have seen very smart people not able to explain at all their work and during a PhD they were forced to improve. I have seen very good presenters that were forced to do some actual work. I have seen people that were convinced they can solve anything (as in some parts of the world the bachelor and master focus to much on solvable problems) understand that sometimes there just isn't a clear, nice solution to a problem.
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.
> 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.
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?
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.
Sure context and audience matters, but even outside of that the article is rather poorly written. This part in particular should really emphasize that she disproved the conjecture, as it stands it almost sounds like she proved it:
> 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.
> "After months of trying to prove the result, I managed to understand why it was so difficult. I realized that if I used that information correctly, I might be able to refute the claim. Finally, after several failed attempts, I found a way to construct a counterexample [a case that does not satisfy the studied property and therefore proves it is not universally true]."
This excerpt features a terrible typo, so I agree it's poorly written, or at least not properly proof read. I don't agree with your specific criticism of the excerpt though. I think the excerpt makes it perfectly clear she disproved the conjecture by highlighting how that was potentially a disappointing outcome.
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.
> 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.
After several hours, the HN title has been modified again to match the original much more closely, from "Hannah Cairo has solved the Mizohata-Takeuchi conjecture"[1] to "Hannah Cairo: 17-year-old teen refutes a math conjecture proposed 40 years ago".
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
Yes, presumably this:
https://terrytao.wordpress.com/2025/02/25/the-three-dimensio...
I will try to incorporate Hannah's notes style into my own stuff; I am using a similar lecturing device with the tablet, but her presentation is soooo far beyond mine. So beautiful!
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.
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-...
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
That's a funny coincidence that the "Radon" there is a person's name and not the radioactive element or its emanations.
Not really related but similar situation and I found it funny when I realized it:
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.
Nominative determinism strikes again.
While extremely talented, I am not surprised to find this coming from a teen. Major mathematical discoveries often have come from those in their mid 20’s with the greater discoveries being skewed towards the younger 20s and teens. I think this because pure mathematics is just so creative.
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.
I wonder how many years on average people can dedicate themselves to think deep about research problems, before they must stop thinking and start managing, or quit the race.
Natural sciences: about 3 years at best.
You finish your degree, and start your PhD. The first year, you are busy learning techniques and getting caught up with the relevant literature. You are far too concentrated on learning new things to get any thinking done.
In your second year of your PhD, you are getting better. You can do most things without thinking about them. This frees up your brain to think about other things. However, your grasp of the wider literature is still lacking, so you use that brainspace to optimise your current experiments (as you should).
In your third year of your PhD, you are starting to write things up: either your thesis, or your first (big) paper. You read a lot more, you know a lot more. The deep thinking can commence.
Your first postdoc is probably your most productive time: you know what you are doing; you know the state of the literature and which parts are reliable and which aren't; you have a clear idea of what problems need solving. You are starting to write your first grant applications, but you only need one for yourself and not several to cover the needs of a full lab. You don't have any kids at home. This is a good time to solve some big problems. It lasts about 2-3 years.
At the start of your second postdoc, you panic. The big problem was harder than you thought and you don't have enough high-impact papers to be competitive in job applications for a principal investigator (PI) role. You start churning out low-value fillers and collaborating with everyone and their hamster to get your name on as many papers as possible. The rest of the time is taken up by applying for grants and PI positions. You don't even make it to the interview stage. You start pondering about life outside of academia.
The big problems are forgotten.
You could replace “research” with nearly any term not undergirded by a direct profit motive (eg civics, politics, education, community health, urban planning). Or maybe it’s just (and you may be saying this) that one can only dedicate themselves up until a clear profit opportunity appears.
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).
> there are numerous examples of productive older mathematicians
Curious about the extreme cases. Did any centenarians ever managed to come with an outstanding original math result? If it didn't happen before, I hope to see it happening in the next decades, given current demographic trends.
I was told that a book published in honor of Oscar Zariski's 80th birthday included a paper by Oscar Zariski, either proving or at least making progress on a longstanding conjecture by Oscar Zariski.
I was in the relevant department at the time (Harvard math), but I wasn't much of an algebraic geometer, so I took that at face value without probing for details.
Erdős? I’m sure the Ritalin helped.
Wiles started working on it at 33 so it took some time to finish it but the base ideas were already in place in his early thirties.
That seems like a pretty weak argument. I don’t think he was just filling in easy details for the next decade plus. Also, even 33 is quite a bit older than the commonly claimed early 20s needed for great mathematical work.
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.
Peter Scholze received his Fields Medal at 30 for work he carried out through his 20’s. However he is not the youngest recipient: J.P. Serre received his Fields medal at 27.
Before we get too excited about the Fields Medal as an indicator for age of great mathematical achievements, let's remember that it's only awarded to people under 40 for work done earlier, potentially many years earlier.
Probably it is because the first solved problem seems like fun, but solving problems daily as a job quickly become boring?
https://en.m.wikipedia.org/wiki/Shoshin
https://en.m.wikipedia.org/wiki/Einstellung_effect
The Fields Medal has a cutoff age of 40 years old.
My understanding is that that’s more about encouraging younger mathematicians rather than an expectation that older ones won’t produce anything worthy.
Sort of. It's complicated and there was politics involved. https://web.archive.org/web/20210324121533/https://www.natur...
Thanks for sharing, it's much more complicated than I believed.
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>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.
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.
The best thing about juniors is they don't know a problem is impossible, so they just do it.
Trying to do anything original and novel in math is extremely hard at any age. to do it at 17 is insanely talented. congrats
Anything original, for that matter.
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https://archive.is/Nr1hH
How often does someone produce work that is normally taught to people who are older than the person who discovered it?
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?
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.
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
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.
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?
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.
Modern PhDs are not designed for people that are smart like this. She's a math savant that obviously has a unique and demonstrably effective way of looking at things, why destroy that with "training how to do research".
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.
This is not true at all. Maths PhDs are excellent for this kind of person, and foster and grow such abilities. No matter how smart you are, training on how to do research is going to improve your capabilities and success.
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.
start on a postdoc :)
> I can’t imagine hiring them as a prof when they’re so young
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.
In math or theoretical fields it’s not unheard of to have young professors. Terence Tao was full professor at 24. Wolfram at 21
She could still do a few years of PhD training and still be a super young prof like those two.
At 17 she’s so young that a uni hiring her would need to think about child labor laws
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.
Noam Elkies too, the youngest ever tenured prof at Harvard. Another parallel he had a pretty famous contradiction proof, of Euler's conjecture. But he didn't find that until age 22, so seems like this girl has a good head start!
theodore kacsynski started teaching at 22. a damn bright spark before he was victimised by the mind control experiment.
Kasynski got his masters degree at 22, which just slightly younger than average.
When she graduates she'll probably be between 20 to 23 years old.
I was one of several math grad students who started at Harvard at age 16 or 17 aroud the same time. Ofer Gabber and Ran Donagi went on to conventional academic math careers. I took a less straightforward career path.
But I was offered an assistant professorship at the Kellogg School of Business at age 21, and have often wondered whether I should perhaps have taken that, or else the research position I was offered at RAND.
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What does someone do with a PhD at age 35? Go into industry? Continue as a postdoc? Open a juice bar? It doesn't matter what she's going to "do" with it. It's accreditation of a certain degree of academic achievement, which she has achieved. Arguing that she doesn't deserve it or needs to earn it the "normal" way is stupid.
Norbert Wiener got his PhD from Harvard when he was 19: https://en.wikipedia.org/wiki/Norbert_Wiener.
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.
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.
PhD by publication usually takes a bit more work. I think they tend to want 3 related papers in a field.
What level or type of publication is required?
They must be peer-reviewed journal papers and I believe they tend to prefer if at least one is well-cited or significant, especially if you have only three papers. It is generally harder to get a PhD by publication than to get a PhD the normal way.
That's a rule of thumb for applied sciences. Plenty of theory PhDs graduate with 1 or 0 papers.
Nobody gets a PhD by publication with 0 publications. This is usually a backdoor for people who have done a lot of work in a field, certainly far more than a PhD thesis, and have just never gotten the credential.
A PhD thesis is itself a publication.
PhDs "by publication" refers to not having to submit work additional that already published to the examining committee.
> Nobody gets a PhD by publication with 0 publications.
Do you have a PhD from a theory department? I do. You're wrong.
Lots of people get PhDs with no publications.
Nobody gets a PhD by publication without publications. It's literally axiomatic.
It's amazing how many people on hn are experts on things they do not have the qualifications to be expert on.
> 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)
Are you aware that "PhD by publication" is a real thing that is a separate path than a normal PhD? It is relatively common for schools in some European countries to offer these, but not that common outside Europe.
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.
>You enroll in the school for ~3 months, write the missing parts, and that's it.
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.
TIL Cambridge University is a degree mill.
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.
That's only available to those with an undergraduate degree from Cambridge who, subsequent to their undergraduate degree, publish work worthy of a Ph.D. and pass a viva.
I'm not sure how common a route ot Ph.D. it is (I never heard of it before), but it sounds like an anachronistic extension of the MA's Oxbridge graduates feel entitled to.
Correct. It is generally an exceptionally rare path to a PhD, but the door is open. I believe I saw the statement that this particular program has granted ~100 degrees over its life compared to ~15000 PhDs from Cambridge.
I agree, you seem to be claiming to be an expert on something.
"PhD by publication" is a specific thing, it's in italics in the previous post.
My Universities offer a "PhD by publication". You basically staple together a bunch of your publications, and write a brief intro. It saves you writing a full PhD document. But, the standard on those publications is quite high -- you certainly wouldn't get one from preprints to arxiv at any University I've ever worked at.
Of course, you can get a PhD with no publications, just write a good PhD. Lots of students do taht.
Did you know that there are universities outside of Europe? And that in those universities, "3 publications plus intro and conclusion for a PhD" is also called (usually) PhD by publication (sometimes it is called a kitchen-sink PhD). And my point was that that rule of thumb does not apply to theory students.
Note the italics please.
So I'm sorry you're right I'm not in expert in Europe's monopoly on the phrase "PhD by publication" but who would want to be an expert in trivial bs like that (answer: apparently you and the other guy!)
Maybe in a few days come back and re-read this thread. Maybe you are having a bad day, I don't know.
You are the one who started talking about people not being experts in having a PhD by publication. No-one else (including me) has bought up that they are an expert on that topic.
A PhD by publication sounds very similar to a higher doctorate (DSc, DLitt, etc). Substantive (as opposed to honoris causa) higher doctorates are awarded based on publication record only. To be eligible for a substantive higher doctorate, you generally are expected to have a PhD first - but it might not be an absolute requirement. You’d generally expect a bunch of papers, but in principle a single publication (if sufficiently groundbreaking) could be enough. While this is very impressive for a 17 year old (I wish I could have done that at 17, or at any other age for that matter), it probably isn’t significant enough for a higher doctorate all by itself. If she’d proved P=NP, different story. (Who knows, maybe she shall-well, probably not, but I’d be very happy to be proven wrong about that.)
A PhD is as much a stamp of endurance as it is a stamp of intelligence or accomplishment.
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 do not think there is "one" purpose for a PhD, and not everyone gets the same thing from it - it depends on what are a person's strong and weak points. I have seen very smart people not able to explain at all their work and during a PhD they were forced to improve. I have seen very good presenters that were forced to do some actual work. I have seen people that were convinced they can solve anything (as in some parts of the world the bachelor and master focus to much on solvable problems) understand that sometimes there just isn't a clear, nice solution to a problem.
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.
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.
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> 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.
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?
>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.
Sure context and audience matters, but even outside of that the article is rather poorly written. This part in particular should really emphasize that she disproved the conjecture, as it stands it almost sounds like she proved it:
> 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.
From the first paragraph of the article:
> "After months of trying to prove the result, I managed to understand why it was so difficult. I realized that if I used that information correctly, I might be able to refute the claim. Finally, after several failed attempts, I found a way to construct a counterexample [a case that does not satisfy the studied property and therefore proves it is not universally true]."
That makes it quite clear that she refuted it.
This excerpt features a terrible typo, so I agree it's poorly written, or at least not properly proof read. I don't agree with your specific criticism of the excerpt though. I think the excerpt makes it perfectly clear she disproved the conjecture by highlighting how that was potentially a disappointing outcome.
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.
> 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.
Anyone have a link to her supposed first paper on number theory? I have my doubt on her counter-example. It uses asymptotic methods rather loosely.
posted 4 days ago https://news.ycombinator.com/item?id=44441730
Original title is more informative than the edited one here.
I submitted under an approximation of the original title, and it was edited within seconds.
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."
[1] https://news.ycombinator.com/newsguidelines.html
After several hours, the HN title has been modified again to match the original much more closely, from "Hannah Cairo has solved the Mizohata-Takeuchi conjecture"[1] to "Hannah Cairo: 17-year-old teen refutes a math conjecture proposed 40 years ago".
[1] https://web.archive.org/web/20250706185220/https://news.ycom...
As the submitter, you can re-edit the title after submission (for some limited time period).
Refuted?
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.
She found more than one way of disproving it in the process.
Yes, found a counterexample to the conjecture.
Great achievement. Now Princeton Math department will ask her to join their school for Ph.D.
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Harmonic analysis?
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Could you please stop posting unsubstantive comments to Hacker News? we're trying for something different here.
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"Ciaro says it required several tools, including fractals, and she had to arrange everything very carefully."
Article could at least spell her name correctly.