“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.”

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.

- 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.

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

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

I wish her the best in her coming career.

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.

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...

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.

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

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.

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.

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.

Amazing story on a no-less amazing teenager.

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

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

”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.

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

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

>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.

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?

> 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!

  – 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

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

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

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

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.

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.