The basis would be getting a dean's approval (or lack of a denial). That is the level at which "Exception Handling" is normally done at a university (above professor, below vice president), and it would normally be the dean of the department that housed the graduate program (some grad programs span multiple departments, which makes it more complicated).
Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).
Every university handles this slightly differently. Exceptions can often be made.
To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.
As a pupil of Dijkstra and seeing at least some rise in formal verification because of the modern tooling and as a follower of Lean (and Agda, Coq, Idris* etc), I hope it will be at least a strive to deliver parts of proofs in code verifiable form. More machine verifiable building blocks will lead to a bettering of everything.
Offtopic, but I am 17 too just like Hannah Cairo but nothing too groundbreaking till now I suppose and it absolutely brings me delight that I can talk to somebody who was a pupil of Dijkstra,
I have heard a lot about dijkstra's algorithm's and I had forgotten about it and so I searched it right now, but the only thing I knew is that it is pretty popular algorithm.
If I had to ask you kind sir, what would be the biggest life lesson (in coding, or anything general) that you could give me be?
I'm on the practical side of things and have dabbled quite a bit with languages like Idris, I think we are mostly far from using them because of ergonomics.
Even in Scala (which is a very advanced language, but still far behind Idris) I deliberately don't use certain type-level features because it increases the compilation times too much (even with incremental compile!). It's very sad, but this is reality.
So, the problem isn't really that we need to "invent airplanes" - we already have them! What we need to do is make them usable and affordable by everyone.
I see more and more languages trying to add type-level features or try to embed other languages like Prolog-like ones into them. I hope that gets traction and becomes ergonomic, otherwise no one will use it in practice.
”Cairo applied to 10 graduate programs. Six rejected her because she didn’t have a college degree. Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions."
This is both unsurprising and shocking to me at the same time.
For institutions of allegedly pure higher learning in a field where it's known that youth is where the advancement happens, the fact 80% axle wrap over a piece of paper that, let's face it, in modern times of grade inflation is pretty much worthless of anything beyond money and sitting in a seat for four years.
Cairo may have a way out: Commonly the main criteria for a Ph.D. is: (1) Pass qualifying exams in several important topics in the field. (2) Do some research that is an "original contribution to knowledge worthy of publication" where the main criteria for publication is "new, correct, and significant".
No professor or university Dean can keep her from doing well on both (1) and (2). It looks like (1) would be easy enough for her. The work she has already done may satisfy (2). Once she has done well on (1) and (2), tough not to award her a Ph.D.
I never got excited for math, thus didn't care much to start with. But then add the issues of high school math and that solidified it for me. Math word problems were the worst. With a dyslexic brain I would consistently read the important words as something different, thus correctly solving the wrong problem and being derided for it.
Geometry required rote memory exceptionalism which also works against my brain design (adhd as well), but let me have the formulas to choose from and I'll get it done. Algebra II? I kept getting in trouble because I would do homework assignments in class instead of paying attention to the teacher trying to teach concepts that were easier for me to learn by reading the examples and following the book. After that, who would ever want to continue beyond the required credits and not think of further math as a masochist hellscape
> Only the University of Maryland and Johns Hopkins University were willing to welcome her straight into a doctoral program. She’ll start at Maryland in the fall. When she finishes, it will be her first degree.
Jeez... what a damning indictment of today's Universities.
She could just use her publication as a dissertation and be done with it!
I see your point, but undergraduate degrees should provide a wide foundation, with little specialization. As you progress to a masters degree, you become more specialized. A doctorate is as specialized as it gets.
It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else. Even to such a degree that they can write a doctorate on it.
But I don't think that's a good excuse to make them forego other curriculum, especially if it is required for other students to take. Schools have a responsibility to educate people to a certain standard, and give them some general breadth.
In my teens I worked with the statistics department at UTMB. That had a cast of characters there; many profs in the 70s and 80s, who'd gotten their degrees before WW2. A number of them had schooling of the form:
Start school at 9-10, do 5 years of public school, got to a 1 year prep, do a year or two of college, do a two year PhD. Most of them had their PhD's by 22.
> She could just use her publication as a dissertation and be done with it!
The purpose of a PhD is not writing a dissertation. It is a research school, and I'm sure she could still learn a thing or two about research (and teaching).
> Jeez... what a damning indictment of today's Universities.
> She could just use her publication as a dissertation and be done with it.
I’m not suggesting this person is doing anything fraudulent as she seems quite impressive.
However, educational institutions get constant requests from parents who want their children to skip far ahead before they’re ready. It’s a competitive world and they know that being able to claim a child skipped several grades or even skipped undergrad entirely is a unique and very impressive achievement for the resume. It also theoretically provides a few additional years of earning potential by giving a career head start.
The first problem is that many of these parents (again, no accusations for this specific case) see this and want to make it happen for their child at any cost. There are some wild stories about parents trying to cheat their kids forward or falsifying their accomplishments to try to skip grades.
The secondary problem is that it can be hard on kids to be thrust forward so far past their peers. I had several friends who skipped a grade in middle school and most of them didn’t have a great experience for social reasons. Skipping undergrad altogether would thrust someone into a foreign world with a lot of baseline expectations and norms that they hadn’t yet learned, combined with no peers their age to discuss it with.
It creates a high chance for burnout or failure, which could leave them worse off than when they started.
That’s why the recommendation is generally to do undergrad at a challenging institution that allows students some upward mobility in specific areas where they’re ahead. No reasonable undergrad program is going to have this person taking Algebra 101, but there are a lot of opportunities for them to jump right into advanced programs and go deep and broad.
I don't see why it would matter. she could quit math now and still be ahead of the majority of mathematician in terms of contributions. The rest is just formalities. I can see her breezing through the undergrad, quals, etc . It would just be a small delay.
There is a story about when Stephen Wolfram applied for a job at AT&T. The HR person asked him where had gone to college. Wolfram confessed that he had never gone to college. Well ok, did you graduate high school? Um no, I'm afraid not. GED? Nope. What about primary school? Wolfram looks more and more more embarrassed, and admits that he didn't even finish primary school. HR person by now is quite uncomfortable when Wolfram manages to think of something. He says "I do happen to have a Ph.D. Does that help?"
“There was this inescapable sameness, in a way. No matter what I did, I was in the same place doing mostly the same things,” she said. “I was very isolated, and nothing I could do could really change that. I’d wake up on certain days and realize, I’m just older.”
I finally have something in common with a math prodigy.
I find the Soviet idea of Math Circles so interesting and important. I bought books on the subject, but it's difficult to implement for your own children only. Nothing beats it like having an actual one, run by math teachers and in your city.
It's wonderful that Khan Academy played a role in enriching her early education. It's proving to be a solid resource across the spectrum of math ability.
There's so many photos of just her staring off into the distance, and only one photo with her presenting the actual thing that she's supposedly famous for. I don't get the point of just all these random photos of this girl.
Not only she did grow in Bulgaria during the most turbolent times of regime change from communism to democracy, but later graduates with a PHD from Harvard, and later becomes Director and Founder of the Berkeley Math Circle, and is also organizer of math competitions in Bay Area, and publisher of what seems to be a complete set of Math Books, carefully crafted with her peers from BG and presented here
I feel like a constructive proof is the best scenario for a young talent like this, they really can use their vivid imagination and manipulate what they are after.
> When I was an undergraduate at MIT I loved it. I thought it was a great place, and I wanted to go to graduate school there too, of course. But when I went to Professor Slater and told him of my intentions, he said, "We won’t let you in here."
> I said, "What?"
> Slater said, "Why do you think you should go to graduate school at MIT?"
> "Because MIT is the best school for science in the country."
> "You think that?"
> "Yeah."
> "That's why you should go to some other school. You should find out how the rest of the world is."
– were all homeschooled. Cairo started learning math using Khan Academy’s online lessons, and she quickly advanced through its standard curriculum. By the time she was 11 years old, she’d finished calculus
No normal childhood and certain kinds of parents, disproportionately more math education, there is certain downsides to all this, that articles like this conveniently avoid to mention.
>Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere.
Rather than speaking about her age or the vague notion of talent, I'd be much more interested in why the rest of the academia was unable to replicate her methods in 40 years or so. From her own admission, it was throwing different ideas and approaches until she saw a disprecancy that ultimately disproved the theorem. There should be many people far more experienced who do have the knowledge to do this, but why did they not? This dosen't look like a intuitive jump, seems more like building a test case that fully stressed the theorem.
45 comments
[ 2.9 ms ] story [ 70.0 ms ] threadI wish her the best in her coming career.
Likely, there are also rule systems that would need to be circumvented; for example, while the admissions program for a graduate program would accept the student, the registrar or other entity might then negate that acceptance due to standing policy. At that point, I wouldn't be surprised if it reached the vice president level of the university (VP of the School that housed the department).
Every university handles this slightly differently. Exceptions can often be made.
To qualify you would normally have to do something truly exceptional or show a pattern of continuously excellent (grad student level). From what I can tell (I am not a math expert) she is considered a 'prodigy' and the problem she solved would require considerable skill, in a way that predicts continued successful performance at the university professor. The history of prodigies who skipped various steps of the normal schooling path is mixed- personally, I think people undervalue the importance of spending time with people in your age group during the 15-25 range, but honestly, without knowing more detail, it's hard to say.
If I had to ask you kind sir, what would be the biggest life lesson (in coding, or anything general) that you could give me be?
Even in Scala (which is a very advanced language, but still far behind Idris) I deliberately don't use certain type-level features because it increases the compilation times too much (even with incremental compile!). It's very sad, but this is reality.
So, the problem isn't really that we need to "invent airplanes" - we already have them! What we need to do is make them usable and affordable by everyone.
I see more and more languages trying to add type-level features or try to embed other languages like Prolog-like ones into them. I hope that gets traction and becomes ergonomic, otherwise no one will use it in practice.
This is both unsurprising and shocking to me at the same time.
For institutions of allegedly pure higher learning in a field where it's known that youth is where the advancement happens, the fact 80% axle wrap over a piece of paper that, let's face it, in modern times of grade inflation is pretty much worthless of anything beyond money and sitting in a seat for four years.
No professor or university Dean can keep her from doing well on both (1) and (2). It looks like (1) would be easy enough for her. The work she has already done may satisfy (2). Once she has done well on (1) and (2), tough not to award her a Ph.D.
Geometry required rote memory exceptionalism which also works against my brain design (adhd as well), but let me have the formulas to choose from and I'll get it done. Algebra II? I kept getting in trouble because I would do homework assignments in class instead of paying attention to the teacher trying to teach concepts that were easier for me to learn by reading the examples and following the book. After that, who would ever want to continue beyond the required credits and not think of further math as a masochist hellscape
Jeez... what a damning indictment of today's Universities.
She could just use her publication as a dissertation and be done with it!
It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else. Even to such a degree that they can write a doctorate on it.
But I don't think that's a good excuse to make them forego other curriculum, especially if it is required for other students to take. Schools have a responsibility to educate people to a certain standard, and give them some general breadth.
I disagree that she should skip the general education.
Prodigy skips undergrad -> Universities are doomed? What?
The purpose of a PhD is not writing a dissertation. It is a research school, and I'm sure she could still learn a thing or two about research (and teaching).
> She could just use her publication as a dissertation and be done with it.
I’m not suggesting this person is doing anything fraudulent as she seems quite impressive.
However, educational institutions get constant requests from parents who want their children to skip far ahead before they’re ready. It’s a competitive world and they know that being able to claim a child skipped several grades or even skipped undergrad entirely is a unique and very impressive achievement for the resume. It also theoretically provides a few additional years of earning potential by giving a career head start.
The first problem is that many of these parents (again, no accusations for this specific case) see this and want to make it happen for their child at any cost. There are some wild stories about parents trying to cheat their kids forward or falsifying their accomplishments to try to skip grades.
The secondary problem is that it can be hard on kids to be thrust forward so far past their peers. I had several friends who skipped a grade in middle school and most of them didn’t have a great experience for social reasons. Skipping undergrad altogether would thrust someone into a foreign world with a lot of baseline expectations and norms that they hadn’t yet learned, combined with no peers their age to discuss it with.
It creates a high chance for burnout or failure, which could leave them worse off than when they started.
That’s why the recommendation is generally to do undergrad at a challenging institution that allows students some upward mobility in specific areas where they’re ahead. No reasonable undergrad program is going to have this person taking Algebra 101, but there are a lot of opportunities for them to jump right into advanced programs and go deep and broad.
- home schooled? check
This on top of her extraordinary talent and hard work. Institutional education truly is a great leveler, at both the top and bottom.
From the article:
https://www.quantamagazine.org/wp-content/uploads/2025/08/Ha...
https://math.berkeley.edu/~stankova/
Not only she did grow in Bulgaria during the most turbolent times of regime change from communism to democracy, but later graduates with a PHD from Harvard, and later becomes Director and Founder of the Berkeley Math Circle, and is also organizer of math competitions in Bay Area, and publisher of what seems to be a complete set of Math Books, carefully crafted with her peers from BG and presented here
https://archi-math.com/
Curious whether miss Cairo was a student of hers or is to be.
Also, I love the handwritten slide on one of the photos. Very nice.
> I said, "What?"
> Slater said, "Why do you think you should go to graduate school at MIT?"
> "Because MIT is the best school for science in the country."
> "You think that?"
> "Yeah."
> "That's why you should go to some other school. You should find out how the rest of the world is."
-- Surely You're Joking Mr. Feynman
Rather than speaking about her age or the vague notion of talent, I'd be much more interested in why the rest of the academia was unable to replicate her methods in 40 years or so. From her own admission, it was throwing different ideas and approaches until she saw a disprecancy that ultimately disproved the theorem. There should be many people far more experienced who do have the knowledge to do this, but why did they not? This dosen't look like a intuitive jump, seems more like building a test case that fully stressed the theorem.