At 17, Hannah Cairo solved a major math mystery

https://www.quantamagazine.org/at-17-hannah-cairo-solved-a-major-math-mystery-20250801/

by baruchel • 12 months ago

474 173 12 months ago

quantamagazine.org

Related: https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)

tocs3• 12 months ago

Earlier discussion: https://news.ycombinator.com/item?id=44481441

I wish her the best in her coming career.

dang• 12 months ago

Thanks! Macroexpanded:

A 17-year-old teen refutes a mathematical conjecture proposed 40 years ago - https://news.ycombinator.com/item?id=44481441 - July 2025 (105 comments)

OutOfHere• 12 months ago

What is the general basis for skipping college and getting admission directly in a Ph.D. program? What does one have to generally do to qualify?

ameliaquining• 12 months ago

AFAIK there's no general procedure; it's just that the admissions committee can admit anyone whom they're convinced can do Ph.D.-level work, and that almost always involves getting an undergraduate degree first but in extremely exceptional cases it might not.

joe_the_user• 12 months ago

Essentially, you have to convince a math professor that you able to complete the Ph.D program. There are a variety of ways to do that. Solving a major open conjuncture is clearly one of them but just getting to know one or another professors can work (I went to school with the semi-famous Paul Lockhart who also did this).

TMWNN• 12 months ago

For context, Stephen Wolfram dropped out of Eton to start at Oxford, then was admitted to Caltech's physics PhD before finishing his Oxford BA, receiving his doctorate at the age of 21.

(He was publishing papers in high school.)

oldpersonintx2• 12 months ago

be actually gifted

this is what gifted looks like, not legions of students who are told they are "gifted" but are just pretty smart

masfuerte• 12 months ago

Yes, but it's not enough.

Be actually gifted and apply to a university that isn't run by dickheads.

pinewurst2• 12 months ago

I knew someone who was admitted to a Masters program at Stanford sans undergraduate degree. His employer at the time was a large donor.

dekhn4• 12 months ago

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.

anonzzzies3• 12 months ago

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.

Imustaskforhelp2• 12 months ago

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?

Timwi• 12 months ago

what would be the biggest life lesson (in coding, or anything general)

Pursue what interests you and what fires up your passion, not what grown-ups tell you is “lucrative”, “prestigious”, “profitable” or somesuch.

lsuresh• 12 months ago

Not the one you asked, but here's what I would have told my 17 year old self.

* "Slope beats y-intercept." The best computer scientists and engineers I've ever worked with and/or mentored embodied this principle more than anything else.

* It can be tempting to over-optimize for short-term milestones (e.g., an important admissions exam, the next job or promotion), but there is a significant compounding value to knowledge accumulation and truly learning your craft well. Read and learn as much as you can, all the time, even if it isn't immediately necessary or useful.

valenterry• 12 months ago

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.

anonzzzies3• 12 months ago

I agree; I would say that most can be taught to people without advanced math/logic degrees/backgrounds, but because the tools are created by (and usually for) people with formal verification majors), they are just quite bad ergonomically as you say. I think we are in a good place of making it these types of proofs easier and faster more mainstream effort, not just one professor and their students, is put behind it. I believe AI will play a part here.

hadlock• 12 months ago

A less click-baity headline might be "17 year old Hannah Cairo has solved the Mizohata-Takeuchi conjecture"

noqc• 12 months ago

probably add "in the negative", to that.

1024core2• 12 months ago

"17 year old Hannah Cairo has disproved the Mizohata-Takeuchi conjecture"

sdenton43• 12 months ago

Is it an important conjecture, or just something someone came up with last week?

cgh2• 12 months ago

This is answered in the article’s second paragraph.

AtlasBarfed• 12 months ago

”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.

kurthr• 12 months ago

A lot of programs don't want to have to babysit a teenager no matter how talented they are. Some of the more prestigious programs both have some experience with it and extra staff to (give profs warm fuzzies) handle any issues that come up. I'd expect it depends a lot on her interests and the particular professors that study that at any particular institution.

Even as a student, I'd be more interested in which professors at Johns Hopkins were accepting students, than which school.

xandrius• 12 months ago

As if that was the reason.

Also, I've seen a great deal of adult babies in academia, so let's not be ageist here.

currymj• 12 months ago

i can understand a university administrator not feeling they can give a good educational experience to a 14 year old.

but she'd be the same age as all the first years on campus, any university must be institutionally equipped to deal with a 17/18 year old. seems like an odd choice to personally intervene to cancel an admission.

like presumably if she enrolled in a bachelor's program, and then signed up for grad math classes while doing independent research, this would be fine. but if she takes the same classes and does the same research for PhD credits instead, no good?

gus_massa• 12 months ago

In my university, when someone is accepted as a graduate student of a different topic (let's say phisics -> biology) it's usual to include a few of the last courses of the major as a mandatory part of the Ph.D.

neilv• 12 months ago

Two admitted her, but then higher-ups in those universities’ administrations overrode those decisions.

I've personally seen universities go both ways, and it comes down both to individuals, and to the culture of the department/university faculty and administration.

(Not to the culture of the student body, which is influenced by the administration culture, but has very little institutional memory, and almost zero power. If you draw an analogy to nations, there might be one with the most awful 'leaders' seizing and abusing power, but that's "way above the pay grade" of the many nice citizens you will meet -- who didn't know what they were being born into, and will do their best to be decent to each other, despite whatever bits they're unfortunate to learn about the upper powers.)

graycat2• 12 months ago

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.

coliveira• 12 months ago

PhDs in the USA also have curricular requirements, with at least some required courses. These courses are only available after a bunch of paperwork is given to the university. So, it's not so easy.

amelius• 12 months ago

What I'd like to know is how many 17 year olds failed to solve their math mystery, and chose a career in programming instead.

geodel• 12 months ago

Almost all of them?

1659447091• 12 months ago

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

kevinventullo• 12 months ago

Many people who succeed in solving their math mystery still end up choosing a career in programming

carabiner• 12 months ago

I hope she doesn't burn out and move into the woods like Grothendieck, Kaczynski,

gspencley• 12 months ago

I have yet to move into the woods but, given my current levels of burnout, I think such a proposition is highly underrated. What's wrong with moving into the woods?

Wait, if everyone moves into the woods there will be no more woods that offer seclusion. Yeah moving into the woods is awful! I hope she doesn't burn out!

bee_rider• 12 months ago

One option could be to get a hammock and go program in a tree. That’s sort of like moving to the woods, but you don’t have to live there full time and you still get to program.

laurent_du• 12 months ago

Kaczynski was really not on the same level. By several orders of magnitude.

BizarroLand• 12 months ago

I mean, force feed her massive doses of LSD and use elite psyops to incessantly harass her as a ̶ ̶p̶r̶a̶n̶k̶ science experiment and maybe

dekhn4• 12 months ago

It's generally well documented that Kacynski was a highly talented mathematician.

1024core2• 12 months ago

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!

rahkiin• 12 months ago

Normally you have a master of science as well. And for that you require a bachelors. So to do it directly you need to do it all at once?

morleytj• 12 months ago

In the US a doctorate usually doesn't require a master's, most people go straight from undergrad to phd.

barry-cotter• 12 months ago

I have a Master’s and never got a Bachelor’s. Universities can admit people more or less as they please, subject only to the very forgiving scrutiny of accreditation agencies or whatever government department officially supervises them.

dragonwriter• 12 months ago

Normally you have a master of science as well. And for that you require a bachelors.

Many Master’s programs require Bachelor’s as a prerequisite for admission, and some of the others include a Bachelor’s at some point, but not all do. (The same is true of the Ph.D. and the stereotypically preceding degrees, and also of some professional degrees and Bachelor’s degrees.)

general1726• 12 months ago

Hey, you are not getting your degree without paying for it.

pinewurst2• 12 months ago

The interesting part is that two others admitted her but retracted after "higher-ups in those universities’ administrations overrode those decisions".

Was she not considered properly conditioned?

MathMonkeyMan2• 12 months ago

"I'm in charge, I enforce the rules, I'm a big deal."

terminalshort2• 12 months ago

Of course she was, but that's not what this is about. Letting her in proves that the bureaucratic credentials offered by schools are meaningless. The university system in its present form in the US is entirely predicated on the fiction that those credentials actually mean something and are worth paying six figures for.

whatshisface• 12 months ago

Two out of ten is pretty good for anything involving individual decisions, just ask any salesperson.

skeptrune5• 12 months ago

I think the biggest flaw with higher education today is that we're pushing people into doing undergraduate degrees who are already well beyond coming out of self learning from high school or other experiences.

jonhohle• 12 months ago

That’s one of the biggest problems? Not pushing people into higher educational programs with little societal or economic value but huge loads of debt?

contravariant• 12 months ago

To the right people a university education can be an asset rather than a barrier to entry.

conorbergin• 12 months ago

Very few people are too “advanced” to be challenged by a sufficiently difficult undergraduate degree, ridiculous thing to say imho, I went to a UK university this decade and I can give you a laundry list of issues more significant than “exceptional students being slowed down”.

Aurornis2• 12 months ago

The number of high schoolers who might be ready to go into postgrad programs is very small.

There is no way this could be the “biggest flaw” in higher education today because the number of people possibly impacted is so tiny.

Although I think you’re striking at something that is a real problem with undergraduate degrees today: Many universities have become so watered down and softened that students spend the first 1-2 years doing what they should have been learning in high school.

My friends who still teach at university constantly complain about students arriving for undergrad with very poor writing, communication, and listening skills.

TheRealPomax• 12 months ago

Conversely: if that's all it takes, there is no point in going to University just to get a piece of paper that says "you did the thing you already did".

University (for folks serious about continuing in academia after) is (obviously) about making sure you have the same base knowledge as everyone else, but also for you to come to terms with how academia actually works, who the bad players are, who the good players are, and who you need to know to get shit to happen for you. So in that sense, most Universities going "no" is literally the most accurate reflection of what life's going to be like on a continuous basis on the inside.

jackero• 12 months ago

50% of my college education was general education though.

And frankly I find it just as valuable.

assword• 12 months ago

Jeez... what a damning indictment of today's Universities.

There’s a modern phenomenon I’ve been thinking about but have struggled to put a name to.

Everything just becomes so generalized, streamlined that it becomes impossible to operate outside of the pre defined “happy path”.

AI will make this increase 100x as taking humans out of the loop seems to accelerate this process.

ninala• 12 months ago

Perhaps the term "canalization" fits? Coined by C.H. Waddington, describing the process of forming a chreode in a homeorhetic system. But it doesn't really encompass the "everything becomes overly generalized" concept you mentioned. Rather, it's more about robustness against perturbation of a trajectory in a conceptual space. Christopher Alexander called them 'paths in configuration space'.

TrackerFF2• 12 months ago

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.

thrawa8387336• 12 months ago

No, that was the purpose of high school. As not practiced in public schools, as not practiced in the US

amradio1989• 12 months ago

I'm not sure higher ed's educational philosophy is serving students all that well. The breadth of education is a shallow survey at best that is quickly forgotten exactly one semester later.

Thankfully the workforce has common sense and will happily snap her up into employment.

sreekotay• 12 months ago

Most of that you can get with diligence in high school and a smattering of College classes as HS junior/senior IMHO.

getnormality• 12 months ago

The woman is not being "made" to do anything. She's already world-class at math and wants to maximize the impact of her specialized talents, so she's going straight to PhD.

I don't think we need to nanny people who break the mold with extraordinary talents to conform to some generic correct educational sequence. They've proven they know how to make something of themselves and their own ideas should count for more.

Izikiel43• 12 months ago

especially if it is required for other students to take

How many of the other students disproved 2 conjectures in advanced calculus that were open for decades?

dvrj101• 12 months ago

It is entirely possible for people to intensely focus on a very, very narrow thing - and ignore everything else.

sounds like Phd

wiredfool• 12 months ago

You say that like it’s a universal thing. It’s much more of a Liberal Arts thing, which is much more prevalent in the US then in Europe.

JAlexoid• 12 months ago

This is ignoring the value of the whole education process, that you go through the years at a university.

I disagree that she should skip the general education.

xp842• 12 months ago

Thinking about my GE requirements at undergrad, I think it would be a waste of this girl's time to be forced to learn about and write about random subjects that don't interest her. She has but one lifetime, and can contribute much to her field.

The subjects such as English composition, she should be allowed to test out of if she is already a good writer.

thechao• 12 months ago

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.

nurettin• 12 months ago

what a damning indictment of today's Universities.

Prodigy skips undergrad -> Universities are doomed? What?

JohnKemeny• 12 months ago

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).

Quekid52• 12 months ago

That's a nice quip, but aren't degrees meant to offer breadth of knowledge? (I'm sure she has lots, but perhaps is weak in other areas.)

Aurornis2• 12 months ago

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.

fn-mote• 12 months ago

Clearly the correct method is: publish your “dissertation” first, then apply for admittance to the PhD program.

throwaway815233• 12 months ago

IIRC Erik Demaine was also homeschooled, skipped college, got a very early PhD, and joined the MIT math faculty as a teenager (he is a full professor there now). Johnny von Neumann OTOH went through a "normal" secondary education partly because his parents wanted him to have traditional social exposure to kids his age while he was growing up. His math training was very accelerated though, and he had professional level research publications at 17.

paulpauper• 12 months ago

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.

throwaway815233• 12 months ago

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?"

conferza• 12 months ago

Wow, this is remarkable. So inspiring to read, even though I'm terrible at mathematics.

impish9208• 12 months ago

    “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.

munificent• 12 months ago

Thank God she found math instead of Factorio.

sbennettmcleish• 12 months ago

the factory must grow ;-)

_benj• 12 months ago

Maybe hot take…

I can see the point of sameness in homeschooling, but compared to traditional education? I’m not sure how much flexibility one would have to teach oneself calculus by 11 or the equivalent of an undergrad in math by 14!

That flexibility must be found in something non-traditional!

I’m no prodigy at all whatsoever but school was mostly dull and filled with teenager drama! Nobody knew what Linux was, cared about music production or anything interesting! The talk was which boy/girl whatever

cultofmetatron• 12 months ago

its so much better nowadays. I"m 40 now and I'm low key jelous of kids today. Today if you want to learn to code, you have freecode camp and chatgpt to ask questions. Math? there's mathacademy and khanacademy. There are so many options now for learning stuff that we didn't have

revskill2• 12 months ago

Loneliness is maththematician best friend.

Xmd5a2• 12 months ago

Straight to jail, right away

A few more months in prison and I would have proved all the great theorems of the time.

André Weil

MathMonkeyMan2• 12 months ago

Here's a link to the paper on the arxiv: https://arxiv.org/abs/2502.06137

shermantanktop2• 12 months ago

- moved between countries or first/second gen immigrant? check

- 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.

stockresearcher• 12 months ago

moved between countries or first/second gen immigrant?

“Cairo grew up in Nassau, the Bahamas, where her parents had moved so that her dad could take a job as a software developer”

“Travel restrictions stranded her family at her grandparents’ house in Chicago. While they were there, she joined the Math Circles of Chicago”

This doesn’t read like an immigrant. It kind of reads like her dad is a fully American finance dev.

coderatlarge• 12 months ago

“After the spring semester ended, her family moved from Davis to Berkeley — her brother had decided to transfer there — and Cairo finally felt able to settle in.”

fascinating that the family follows the kids’ educational steps.

gosub100• 12 months ago

I wonder what would have happened if she said "mom, dad, I want to be a fashion designer/chef/interior decorator." Undoubtedly, they would have supported her ambition and individuality..of course.

fmbb• 12 months ago

Are home schooled over-represented among mathematicians that solved major math mysteries?

Wait that would not even prove anything.

Are there no mathematicians who solved major math mysteries that went to school?

shermantanktop2• 12 months ago

I don’t know about major math achievements- the n is pretty small- but homeschooled kids are overrepresented in geography bees, spelling bees, and other academic competitions at the high school level.

https://readlion.com/homeschoolers-dominate-inland-northwest...

It appears that homeschooling enables students to pursue singular passions further than students in schools. It’s not for everyone though.

My point was just to note some other interesting characteristics beyond the age and gender that show up in coverage.

zahlman• 12 months ago

Previously: Hannah Cairo: 17-year-old teen refutes a math conjecture proposed 40 years ago (https://news.ycombinator.com/item?id=44481441)

fnord77• 12 months ago

Wait, what software engineering jobs require you to move to the Bahamas?

cgh2• 12 months ago

In my experience, online gambling or banking.

smithkl42• 12 months ago

Insurance is a pretty common one. Lots of insurance companies have offices there - and if you're a top-notch dev, you can make a crap-ton of money as well.

OldfieldFund• 12 months ago

Rich people offshore money there

myth_drannon• 12 months ago

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.

jarvist• 12 months ago

Absolutely. Key for me was to invite children's friends (and family) along, host it in our house and make it a recurring weekly thing. The books (presumably you also have these from the MSRI's 'Mathematical Circles Library') are great, but week-to-week I've found the free online NRICH resources much more directly useful: https://nrich.maths.org/about-nrich

zavg• 12 months ago

This is the most impressive thing I've seen in years.

AlanYx• 12 months ago

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.

spiderxxxx• 12 months ago

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.

azornathogron• 12 months ago

It's not very easy to take photos of mathematics, and the girl is the central subject of the article anyway. What images would you expect to see?

debo_• 12 months ago

Her notes are so clear and so artfully wrought! I wonder if learning from online resources makes one naturally focus more on presentation.

From the article:

https://www.quantamagazine.org/wp-content/uploads/2025/08/Ha...

jagged-chisel2• 12 months ago

Looks like a prepared presentation rather than notes.

baq• 12 months ago

If all presentations were prepared like this I’d probably attend them

thrance3• 12 months ago

I studied math and knew a girl like that, spending an inordinate amount of times making sure to understand every single details of what we were taught, even going further when I could barely keep up. She would also make those pretty, colorful notes with visual interpretations and drawings in the margin. It takes a special form of love for the discipline. She just finished her PhD and is now starting a postdoctoral contract.

larodi• 12 months ago

Zvezdalina Stankova who comments on miss Cairo is on her own super out of the ordinary.

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.

throwaway815233• 12 months ago

Alexander Givental is also involved with that, I think.

https://sumizdat.com/homepage/index.html scroll down.

rossant• 12 months ago

Amazing story on a no-less amazing teenager.

Also, I love the handwritten slide on one of the photos. Very nice.

shusaku• 12 months ago

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.

yumraj• 12 months ago

Wonder why Berkeley didn’t offer her PhD admission given that she was already working with the Prof there.

terminalbraid• 12 months ago

I've been at institutions that explicitly discourage that because of that reason. They want you to get breadth of experience rather than work with the same one person for your early career.

IvyMike• 12 months ago

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."

-- Surely You're Joking Mr. Feynman

diimdeep• 12 months ago

  – 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.

thrance3• 12 months ago

    “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 don't think they're omitting it, but yeah. A lot of homeschooled kids don't fare nearly as well.

corimaith• 12 months ago

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.

thrance3• 12 months ago

Heh, take a look at her paper [0]. It is much more complex that TFA makes it out to be: once you have the intuition you need to prove it actually disproves the theorem.

Also, it's probably a not very well known conjecture that people wouldn't care too much about. Also also, it was a conjecture: people were probably looking to prove it rather that disprove it, failing and then giving up.

[0] https://arxiv.org/abs/2502.06137

PilotJeff• 12 months ago

This math stuff is corrupting our youth.

Besides how can she just skip college and go right into a PhD? She hasn’t even taken intro to Sociology or poetry yet.

At 17, Hannah Cairo solved a major math mystery

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