17-year-old refutes math conjecture, enters PhD program without a degree

Miscellaneous and Personal Stuff I Must Share
1

I’ve read so many of these articles that turned out to be fluffery and bullshit that I almost didn’t look at this, but it’s a fascinating story:

At 17, Hannah Cairo Solved a Major Math Mystery | Quanta Magazine

Yet a paper posted on February 10(opens a new tab) left the math world by turns stunned, delighted and ready to welcome a bold new talent into its midst. Its author was Hannah Cairo(opens a new tab), just 17 at the time. She had solved a 40-year-old mystery about how functions behave, called the Mizohata-Takeuchi conjecture.

“We were all shocked, absolutely. I don’t remember ever seeing anything like that,” said Itamar Oliveira(opens a new tab) of the University of Birmingham, who has spent the past two years trying to prove that the conjecture was true. In her paper, Cairo showed that it’s false. The result defies mathematicians’ usual intuitions about what functions can and cannot do.

The math world is also adjusting to the fact of Cairo herself. After completing the proof, she decided to apply straight to graduate school, skipping college (and a high school diploma) altogether. As she saw it, she was already living the life of a graduate student. 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.

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.

2

Very interesting story.

3

The conjecture is obscure (read “advanced”) enough that the only mention of it on wiki is a brief article about Ms. Cairo and her refutation.

I was hoping to read up enough to have a some vague notion of what was being discussed, but I think that’s a forlorn hope.

4

This is a good read too. Ope I see it was the first link in the OP.

5

The cited article actually contains an attempt to describe the conjecture and Cairo’s proof in layman’s terms. It’s fairly far down in the text.

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From a “technical” side, is it smart to go right to a graduate degree without learning the underlying math of college and MS-level? Either such a person would have no biases based on “old math”, or on the other hand you run the risk of not knowing something basic.

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She already learned the underlying math on her own. What she is doing is completely appropriate.

8

If she already knows everything why go for a doctorate? just start solving problems free lance.

9

Are you being serious here?

Because she wants to work in academia with other people doing similar things. If you actually read the article and understood it, that’s what she was already doing and she loved it and she wanted to do it formally.

10

Yes, she had already mastered calculus at the age of 11, when most kids would be in sixth grade. IIRC, I didn’t encounter calculus until the first year of college.

Believe it or not, she doesn’t “know everything”. :roll_eyes:

But she probably knows way more math than a college graduate with a science degree.

11

How some escape by reading books or gaming she escaped the dullness with mathematics.

But Cairo found homeschooling confining.

“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.”

Math became a kind of escape, a space that felt expansive when her daily life was not.

“Mathematics was another world I could explore. A world that was not confining, a world I could access at any point just by thinking about it,” she said. “That’s how I grew up, thinking about mathematics as this world of ideas that I can explore on my own. That sort of process helped me see math differently than a lot of people.”

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I also had some questions about the wisdom of this choice, but for a different reason: most academic careers are going to involve some amount of teaching students who are not at the same ability level as Hannah clearly is, and given her homeschooled background, it doesn’t look like she has ever really had the experience of sitting in a classroom with students who are basixally “ordinary,” or of being one of those ordinary students in a subject she isn’t especially gifted at. Both of these are valuable experiences if you are planning on any sort of teaching career! Now, it may be that she’ll luck into a job with no teaching responsibilities, or one that only involves working with advanced grad students, but those are pretty rare, and if I were on the hiring committee for any other sort of academic job, I would likely see that background as a red flag unless the candidate had a very strong, demonstrated ability to teach an undergraduate gen ed class without losing their mind or leaving their students hopelessly confused.

13

She apparently knows enough to skip HS, graduate and post graduate school.

If those fine degrees offer nothing, then again, why a doctorate degree? It’s just a piece of paper. Just start doing maths with those poor doctorate students and phds who insisted on doing things the old fashioned way.

I mean, everybody has a masters!

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Not really all that rare for professors with exceptional credentials, which I’m sure Cairo would be if she decides on a career in academia. Such profs tend to spend their time doing research and mentoring graduate students.

No one ever said that.

You miss the point. A doctoral program is not about the degree, it’s about the opportunity to do significant new research under the leadership of a mentor.

15

Ok you’re just being willfully ignorant at this point. She’s doing it that way because, and try hard to wrap your mind around this, that’s what she fucking wants to do. Considering that she’s a super genius and you’re you, maybe she’s on the right track.

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Fair enough – I may be looking at this through a humanities-job-market lens, where research / graduate-teaching-only jobs are quite rare and not the kind of thing anyone can count on getting, regardless of their background, and what constitutes “exceptional credentials” is less clear-cut.

17

You are wondering about the wisdom of taking a permanent position where one is free to do research, organize seminars, and work with colleagues/postdocs/students as one sees fit versus one where you have to teach two or three “Calculus for Engineers” every semester? It is pretty nice, if you can get it.

Your responsibility to avoid the pitfalls, as well:

When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don’t get any ideas for a while: They have every opportunity to do something, and they’re not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come. Nothing happens because there’s not enough real activity and challenge: You’re not in contact with the experimental guys. You don’t have to think how to answer questions from the students. Nothing!

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Even nicer if, as in an example of which I have first-hand knowledge, the school says “you have no undergraduate teaching duties, we want you to focus on research, and BTW, here is a nice new research facility we set up just for you and your grad students to do it in”. It’s amazing what you can get when you have a suitable reputation.

I don’t know what this guy is complaining about, but I very much doubt that teaching undergraduate courses would do anything to inspire better research. Inspiration generally comes from colleagues and the peer community.

If this guy’s ruminations were right, Princeton’s Institute for Advanced Study would not have produced 36 Nobel Laureates, 46 of the 64 Fields Medalists, and 24 of the 28 Abel Prize Laureates, as well as many MacArthur Fellows and Wolf Prize winners. Likewise, IBM Research has found it productive and profitable to designate their top scientists as “IBM Fellows”, which basically means “from now on, your job responsibility is to do whatever you want”.

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Not just IBM. I have worked for a few tech companies that produced cutting edge products that required new science. All of them had a handful of those people. I was way on the other end doing manufacturing but occasionally they’d need my assistance. These people were all stratospherically intelligent and earned their position.

20

More anecdotes coming, but I think they apply. My father taught until retirement at NMSU, my mother all taught at both the university and lower levels, and my wife taught undergrad classes (both at her school and the local community college) while doing her Master’s and PhD.

All of them, in 3 different fields (only my wife is STEM though) said in all the universities they worked with (7 between the three of them!) there were teachers there to teach because they loved it (all three in this group to various degrees), because it was needed to secure the funding they wanted for their own research (my mother and my wife to a degree), or a chore they put up with and skeeved off as much as possible from because it was a requirement for tenure/grants/lab space (none of them).

Too the point that my father in the first category said he was going to quit and go back to business (which he’s done several times) if NMSU (where he worked when I was a child through my undergrad degree) if they made him publish papers and the like. And they caved.

My mother and wife both liked teaching, but my mom wanted more time to research, and my wife found writing papers a waste of time that could be done doing more practical things, not to mention the PITA condescending crap that my wife got trying to publish papers in STEM as a woman (and this is from just 3-5 years ago, not decades ago!!!).

But they all spoke of the many, many examples of people who were there for their own research, personal passions, or just the support framework of the university and bemoaned having to actually teach classes.

Evidently most schools have a number in each of these three broad groups, and depending on what the school wants to accomplish (or afford!) it can support all types, focus on one or two of the types above, or go all in on practical (teaching) or research priorities. Sounds like Hannah Cairo may well be going the last route, and I expect it would be good for her.

Plus all teachers I’ve met have areas they’re good at AND bad at. I know some who love teaching, but aren’t that great at it. And some who loathe it as a chore, but despite that are EXCELLENT teachers. Ms. Cairo is young enough that she knows what she’s passionate (and obviously) skilled at - mixing with like minds while learning more is going to give her time to learn what secondary skills she may enjoy or be good at (including teaching) and if she does/does not enjoy them.

Not (NOT) dismissing her based on her age. She obviously wants to learn more and grow her knowledge, skills, and abilities beyond the restrictions she’s had so far in her life. Power to her!