[HN Gopher] At 17, Hannah Cairo solved a major math mystery
       ___________________________________________________________________
        
       At 17, Hannah Cairo solved a major math mystery
        
       Author : baruchel
       Score  : 176 points
       Date   : 2025-08-01 16:35 UTC (6 hours ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | tocs3 wrote:
       | Earlier discussion: https://news.ycombinator.com/item?id=44481441
       | 
       | I wish her the best in her coming career.
        
         | dang wrote:
         | Thanks! Macroexpanded:
         | 
         |  _Hannah Cairo: 17-year-old teen refutes a math conjecture
         | proposed 40 years ago_ -
         | https://news.ycombinator.com/item?id=44481441 - July 2025 (105
         | comments)
        
       | OutOfHere wrote:
       | 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 wrote:
         | 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 wrote:
         | 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 wrote:
         | 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 wrote:
         | be actually gifted
         | 
         | this is what gifted looks like, not legions of students who are
         | told they are "gifted" but are just pretty smart
        
         | pinewurst wrote:
         | I knew someone who was admitted to a Masters program at
         | Stanford sans undergraduate degree. His employer at the time
         | was a large donor.
        
       | anonzzzies wrote:
       | 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.
        
         | Imustaskforhelp wrote:
         | 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?
        
       | hadlock wrote:
       | A less click-baity headline might be "17 year old Hannah Cairo
       | has solved the Mizohata-Takeuchi conjecture"
        
         | noqc wrote:
         | probably add "in the negative", to that.
        
           | 1024core wrote:
           | "17 year old Hannah Cairo has disproved the Mizohata-Takeuchi
           | conjecture"
        
         | sdenton4 wrote:
         | Is it an important conjecture, or just something someone came
         | up with last week?
        
           | cgh wrote:
           | This is answered in the article's second paragraph.
        
             | sdenton4 wrote:
             | But I don't know that from the proposed re-titling of the
             | article. The importance of the conjecture is more salient
             | for the general reader than the (uninformative) names of
             | the people who conjectured in the first place.
        
         | spiderxxxx wrote:
         | 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 wrote:
           | 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?
        
       | AtlasBarfed wrote:
       | "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 wrote:
         | 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 wrote:
           | 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.
        
         | gus_massa wrote:
         | 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 wrote:
         | > _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.)
        
       | amelius wrote:
       | 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 wrote:
         | Almost all of them?
        
         | 1659447091 wrote:
         | 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 wrote:
         | Many people who succeed in solving their math mystery still end
         | up choosing a career in programming
        
       | carabiner wrote:
       | I hope she doesn't burn out and move into the woods like
       | Grothendieck, Kaczynski,
        
         | gspencley wrote:
         | 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 wrote:
           | 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 wrote:
         | Kaczynski was really not on the same level. By several orders
         | of magnitude.
        
           | BizarroLand wrote:
           | I mean, force feed her massive doses of LSD and use elite
           | psyops to incessantly harass her as a  prank science
           | experiment and maybe
        
       | 1024core wrote:
       | > 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 wrote:
         | 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 wrote:
           | In the US a doctorate usually doesn't require a master's,
           | most people go straight from undergrad to phd.
        
             | sdenton4 wrote:
             | To add slightly more detail: Most research math programs in
             | the US are 5(ish) year 'combined' masters+phd programs. The
             | first couple years are basically course work, seminars,
             | finding your area and advisor, and then the rest is the
             | actual research work. It's not uncommon to leave after the
             | first couple years with a Master's degree.
        
         | general1726 wrote:
         | Hey, you are not getting your degree without paying for it.
        
         | pinewurst wrote:
         | 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?
        
           | MathMonkeyMan wrote:
           | "I'm in charge, I enforce the rules, I'm a big deal."
        
           | terminalshort wrote:
           | 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 wrote:
         | Two out of ten is pretty good for anything involving individual
         | decisions, just ask any salesperson.
        
         | skeptrune wrote:
         | 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 wrote:
           | 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?
        
             | skeptrune wrote:
             | I think research and higher education does have value for
             | the most part. It's undergrad that's really worthless and
             | something people only do for the experience.
        
               | xp84 wrote:
               | Indeed. I've long felt that most undergrad students would
               | be better served by a typical college minus the formal
               | classes. Basically dorms and all the other amenities
               | found in a typical college campus, where you mainly gain
               | life skills and mingle with other people your age.
               | Because most people I met at an average 4-year school
               | were there because it's a societal expectation among
               | certain classes, it's less scary than just getting a job
               | and figuring out life completely on your own, and it is
               | 10x-100x easier to make friends at college than just "out
               | in the world." Not on the list: to learn from college
               | classes, which at an average school teach you less than
               | you'd get from a $200 a year subscription to Great
               | Courses Plus or Brilliant. Or free from Khan Academy.
               | 
               | I know a few very special schools give undergrads access
               | to brilliant minds in their field, but I also have been
               | told that undergrads at those schools are mostly taught
               | by grad students, so I'm not sure that Ivies provide a
               | lot either, beyond the opportunity to hobnob with the
               | legacies that will be running Goldman Sachs in 20 years.
        
               | psyklic wrote:
               | Undergrads who care about learning and research will take
               | the most challenging classes, do research with
               | professors, and surround themselves with other strong
               | students who will push them.
               | 
               | Even at top universities, very very few freshmen are
               | capable of doing high-quality research immediately.
               | They'd be better served learning the foundations inside
               | and out with a cohort of similarly strong students to
               | challenge them.
        
               | skeptrune wrote:
               | If you do research during your 4 year undergrad. You
               | shouldn't have been undergrad. It's really that simple.
        
               | psyklic wrote:
               | I'm not sure that's a simple argument and can't imagine
               | many would agree.
               | 
               | Undergrads who do research generally aren't very good at
               | research yet. A major reason is they either lack or don't
               | fully understand the pre-reqs, which they progressively
               | and cumulatively learn during undergrad. A student can be
               | incredibly smart, but acquiring a strong rigorous math
               | background will still take years.
        
               | skeptrune wrote:
               | Working on that skill and ability is the entire point of
               | postgrad. If those are the skills you're working on then
               | you should be in a postgrad program.
        
               | psyklic wrote:
               | If you don't know the foundations well, you don't belong
               | in a postgrad program. That's the reality and how it
               | currently works. Undergrad teaches you those foundations.
               | 
               | Anyone can try doing research, even undergrads who half-
               | know the foundations. However, trying research doesn't
               | mean you have the background to do great research or to
               | succeed in a postgrad program.
        
               | skeptrune wrote:
               | Let me ammend my statement. *"Anyone who succeeds at
               | publishing research deserves to be in a postgrad
               | program."
               | 
               | Plenty of people in postgrad programs don't know the
               | foundations. It's ok. You are there to learn.
               | 
               | Completely unfair to expect someone already doing
               | research to slog out 4yrs of classes not furthering their
               | career.
        
               | dh2022 wrote:
               | About pre-reqs: third and fourth year PureMath classes at
               | UofWaterloo consisted of math I already took in
               | HighSchool in Romania: group theory, ring theory. Plus
               | some calculus I already read in high school out of
               | curiosity: measure theory and the Lebesgue integral.
               | Another Romanian guy at UofW was auditing 4th year
               | classes while in his first year (he is now a math
               | professor at an American university)
               | 
               | I can see a committed and gifted student being able to
               | get most of the pre-reqs for doctoral studies in America
               | or Canada while in high school.
        
               | Quekid5 wrote:
               | People sometimes _accidentally_ do research. I 'm not
               | joking.
        
               | cge wrote:
               | To agree with you: I've worked with several really
               | brilliant undergrads doing and publishing great research.
               | But all of them were rightfully undergrads. Even if they
               | were actually capable of doing great research, they
               | benefited from the breadth.
               | 
               | If you have bright enough undergrads, you change the
               | curriculum for them within their field of expertise, so
               | that they still get the breadth of things outside it
               | while not wasting time with things they know. You let
               | them not take as many classes, take graduate courses, do
               | more research, take more courses from other departments
               | in related areas but with different perspectives, and so
               | on.
               | 
               | When I was an undergrad, in physics, there was a
               | professor in the department who had done his undergrad
               | there and was legendary, as was quietly mentioned in awe,
               | for not taking any undergraduate physics courses while
               | there; the department had let him skip _all_ of them, and
               | instead take graduate courses and do research.
        
             | franga2000 wrote:
             | The debt part is just a US thing, but the rest of us still
             | have the other problems.
        
               | globalnode wrote:
               | australia copied the US in that regard. they used to say
               | we were the 51st state but i dont even think we qualify
               | as that... a territory perhaps?
        
           | contravariant wrote:
           | To the right people a university education can be an asset
           | rather than a barrier to entry.
        
           | conorbergin wrote:
           | 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".
        
             | TheOtherHobbes wrote:
             | This woman is undoubtedly exceptional. But we don't know
             | how exceptional, because she's an outlier, educated using
             | different methods.
             | 
             | We have no idea how many other people would achieve
             | something similar with a similar background. Personally I'd
             | bet almost anything it's a larger number than most people
             | expect.
             | 
             | I'd also be surprised if she doesn't already have a pretty
             | solid background in undergrad-level math.
             | 
             | The irony is she's actually more typical than not.
             | Universities in the past were open to giving unusual
             | talents special treatment.
             | 
             | Historically, the idea that everyone _must_ follow the same
             | path on the same timetable is unusual.
        
           | Aurornis wrote:
           | 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 wrote:
         | 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 wrote:
         | 50% of my college education was general education though.
         | 
         | And frankly I find it just as valuable.
        
         | assword wrote:
         | > 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 wrote:
           | 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'.
        
             | willhslade wrote:
             | Object orientation applied to life.
        
         | TrackerFF wrote:
         | 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 wrote:
           | No, that was the purpose of high school. As not practiced in
           | public schools, as not practiced in the US
        
           | amradio1989 wrote:
           | 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.
        
         | JAlexoid wrote:
         | 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.
        
           | xp84 wrote:
           | 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.
        
             | com2kid wrote:
             | > 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.
             | 
             | This is a common STEM view, but it is inherently wrong.
             | IMHO it dates back to the unfortunate divide between
             | science and the liberal arts, whereas both were once
             | considered a single field, now days there is disdain and
             | mistrust between the two sides.
             | 
             | The point of history classes isn't to memorize the dates of
             | wars, is it to understand the motivations of humans, it is
             | to understand how the world we lived in has been shaped
             | throughout time, and it is to learn how to do, and
             | understand, research about the history of a place.
             | 
             | The point of English classes isn't just to get good at
             | writing, it is to get good at various types of writing, it
             | is to learn how to read different forms of literature, and
             | it is to have a guided tour through a chosen selection of
             | literature to hopefully develop one's character and
             | thoughtfulness.
             | 
             | One of the most valuable classes I ever took at University
             | was the Art Of Listening To Music. We started off around
             | 500ad or so and went forward through time up until about
             | 1920. We learned the vocabulary of music, how to sit down
             | and listen to a piece of music and describe what we were
             | hearing. After I was done with the class I went from
             | appreciating a handfuls of genres of music to appreciating
             | music itself no matter the genre. It was a 3 credit
             | guaranteed A class that had enriched my life by an enormous
             | amount.
             | 
             | If you really love your major, then the point of going to
             | university was _NOT_ your major, odds are you would 've
             | studied that field with or without the school. (Barring
             | fields that require large capital investments, chemistry,
             | physics, playing with an entire orchestra, building
             | airplanes, etc) The point is _everything else_.
        
               | terminalshort wrote:
               | But that isn't how it works in reality, at least in the
               | US. In reality, outside of their major (and sometimes
               | inside it too), students usually pick the absolute
               | easiest classes that satisfy the requirements. The ones
               | that are known that the teacher doesn't take attendance
               | are heavily desired. And the university is happy to
               | oblige. Departments are funded based on a formula of how
               | many students are in their classes, and they know that if
               | they gain a reputation for being hard, students won't
               | take their classes for GE requirements. It's a race to
               | the bottom. So most departments offer enormous 1 level
               | classes with 200 students taught with minimal rigor, and
               | where you really only have to study a few days before the
               | final to collect your A. And on top of that the frats all
               | keep collections of graded tests from every class for
               | years past, so basically anyone who wants to cheat is
               | able to do so easily. This isn't education. This isn't
               | worth six figures.
        
               | com2kid wrote:
               | > In reality, outside of their major (and sometimes
               | inside it too), students usually pick the absolute
               | easiest classes that satisfy the requirements.
               | 
               | In reality, no one eats healthy food, everyone eats fast
               | food hamburgers all day. Just look at the sales numbers
               | of fast food / junk food VS organic lunch salad bars!
               | 
               | Except, that isn't true. Some people eat junk food all
               | day, and some people choose to eat healthy. Obviously in
               | America we have a bit of a bias, but just looking at
               | averages doesn't give a complete picture.
               | 
               | > The ones that are known that the teacher doesn't take
               | attendance are heavily desired.
               | 
               | Almost none of my university classes took attendance. Why
               | would they? We were paying to be there, if we wanted to
               | waste our money, it wasn't the university's problem.
               | 
               | > So most departments offer enormous 1 level classes with
               | 200 students taught with minimal rigor
               | 
               | Reading books and writing essay's doesn't require rigor,
               | the learning is in the doing. I put in honest work to
               | learn and I got honest feedback from my 100 and 200 level
               | professors, which was all I expected.
               | 
               | > And on top of that the frats all keep collections of
               | graded tests from every class for years past
               | 
               | Almost none of my GE classes used multiple choice tests.
               | They were typically essay tests, written in class.
               | 
               | I should note I did my GE requirements at a local
               | community college, where class sizes averaged ~20-30
               | students, professors had office hours, and I think I only
               | saw a TA once.
               | 
               | > This isn't education. This isn't worth six figures.
               | 
               | The price is too high yes, but a university is place to
               | go where you can dedicate yourself fully to learning,
               | hopefully w/o other outside worries (sky high tuition
               | ruins that...). What a person chooses to do with that
               | time is up to them.
               | 
               | Now one can argue that the worth of a degree is lessened
               | by students who didn't actually learn all that much also
               | being in possession of one. That is a closely related,
               | but separate topic.
               | 
               | That said, the poster I was originally replying to was
               | indirectly advocating for not caring about one's GE
               | classes, and I was replying that one should indeed care,
               | because those classes are incredibly important!
        
         | thechao wrote:
         | 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 wrote:
         | > what a damning indictment of today's Universities.
         | 
         | Prodigy skips undergrad -> Universities are doomed? What?
        
         | JohnKemeny wrote:
         | > _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).
        
         | Quekid5 wrote:
         | 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.)
        
         | Aurornis wrote:
         | > 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 wrote:
           | Clearly the correct method is: publish your "dissertation"
           | first, then apply for admittance to the PhD program.
        
           | throwaway81523 wrote:
           | 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.
        
       | conferza wrote:
       | Wow, this is remarkable. So inspiring to read, even though I'm
       | terrible at mathematics.
        
       | impish9208 wrote:
       | "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 wrote:
         | Thank God she found math instead of Factorio.
        
       | MathMonkeyMan wrote:
       | Here's a link to the paper on the arxiv:
       | https://arxiv.org/abs/2502.06137
        
       | shermantanktop wrote:
       | - 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.
        
       | zahlman wrote:
       | Previously: Hannah Cairo: 17-year-old teen refutes a math
       | conjecture proposed 40 years ago
       | (https://news.ycombinator.com/item?id=44481441)
        
       | fnord77 wrote:
       | Wait, what software engineering jobs require you to move to the
       | Bahamas?
        
         | cgh wrote:
         | In my experience, online gambling or banking.
        
         | smithkl42 wrote:
         | 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.
        
       | myth_drannon wrote:
       | 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.
        
       | zavg wrote:
       | This is the most impressive thing I've seen in years.
        
       | AlanYx wrote:
       | 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.
        
       | debo_ wrote:
       | 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...
        
       | larodi wrote:
       | 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.
        
       | rossant wrote:
       | Amazing story on a no-less amazing teenager.
       | 
       | Also, I love the handwritten slide on one of the photos. Very
       | nice.
        
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