Latest news with #HannahCairo
News18
6 days ago
- Science
- News18
Homeschooled Student Solves 40-Year-Old Math Theory, Gets Direct PhD Admission
Last Updated: Hannah has been offered a place in the University of Maryland's PhD programme in mathematics, bypassing the need for a high school diploma or college degree. Anyone can earn a PhD if they complete their degree and postgraduate studies, pass difficult entrance exams, and qualify at the top. However, a young woman who studied at home rather than attending school got the chance to pursue a PhD directly after solving a mathematical theory that had been a mystery for 40 years. Seventeen-year-old Hannah Cairo from the Bahamas achieved this remarkable feat. Hannah disproved the Mizohata–Takeuchi conjecture, a longstanding problem in mathematics within the realm of harmonic analysis, which studies how waves behave on curved surfaces. Many mathematicians had attempted to solve it over the years, but Hannah proved it wrong and provided a clear example to demonstrate why. Born in the Bahamas and homeschooled by her parents, who encouraged her to delve deeply into subjects of interest, Hannah chose mathematics. By age 11, she had mastered calculus and went on to study university-level subjects such as linear algebra, differential equations, and topology. She read books, took online classes, and occasionally had tutoring. During the COVID-19 lockdowns, Hannah joined the Chicago Math Circle online, where she engaged with problems more advanced than typical schoolwork. This introduced her to the questions and ideas that professional mathematicians grapple with. At 14, Hannah applied to the Berkeley Math Circle summer programme, stating on her application that she had already covered the advanced undergraduate math curriculum—a true claim. Her application was accepted, allowing her to join Berkeley's concurrent enrolment program and take university-level classes alongside graduate students, all while still officially in school. A professor named Ruixiang Zhang introduced Hannah to the Mizohata-Takeuchi conjecture, which predicted limits on how waves can focus on curved surfaces. While many students might have viewed it as a challenging puzzle, Hannah sought to create examples that defied the conjecture's rules, despite frequently encountering dead ends. Although Zhang doubted some of her ideas, Hannah persisted. Her breakthrough came with a seemingly simple idea: instead of trying to work within the pattern of the conjecture, she thought of creating an entirely different wave pattern. She developed waves that extended rather than intersected. After thorough testing, her example showed that the conjecture's predictions were completely incorrect. In February, Hannah published her research paper on arXiv. Some experts were astonished and congratulated her, while others were sceptical, questioning how someone so young could achieve such a feat. Consequently, Hannah was offered a place in the University of Maryland's PhD programme in mathematics, bypassing the need for a high school diploma or college degree. view comments First Published: August 13, 2025, 14:36 IST Disclaimer: Comments reflect users' views, not News18's. Please keep discussions respectful and constructive. Abusive, defamatory, or illegal comments will be removed. News18 may disable any comment at its discretion. By posting, you agree to our Terms of Use and Privacy Policy.
Yahoo
10-08-2025
- Science
- Yahoo
How Teen Mathematician Hannah Cairo Disproved a Major Conjecture in Harmonic Analysis
When she was just 17 years old, Hannah Cairo disproved the Mizohata-Takeuchi conjecture, breaking a four-decade-old mathematical assumption When Hannah Cairo was 17 years old, she disproved the Mizohata-Takeuchi conjecture, a long-standing guess in the field of harmonic analysis about how waves behave on curved surfaces. The conjecture was posed in the 1980s, and mathematicians had been trying to prove it ever since. If the Mizohata-Takeuchi conjecture turned out to be true, it would illuminate many other significant questions in the field. But after hitting wall after wall trying to prove it, Cairo managed to come up with a counterexample: a circumstance where the waves don't behave as predicted by the conjecture. Therefore, the conjecture can't be true. Cairo got hooked on the problem after being assigned a simpler version of the conjecture to prove as a homework assignment for a class she was taking at the University of California, Berkeley. 'It took me a while to convince [course instructor] Ruixiang Zhang that my proposal was actually correct,' she says. Now, under Zhang's advisement, she has a paper on the preprint server and was invited to present her results at the International Conference on Harmonic Analysis and Partial Differential Equations in El Escorial, Spain. Cairo says she loves talking about her research and giving presentations with colorful and descriptive slides (see examples below). When asked what she studies, Cairo says, in short, 'points, lines and waves.' Born and raised in the Bahamas, Cairo moved to California at the age of 16, where she began to take classes at U.C. Berkeley. Now, at 18 years old, she is on to a Ph.D. program at the University of Maryland to continue her research in Fourier restriction theory. Cairo has faced many difficulties in her journey, but she has found comfort and belonging in the field of mathematics and in the work itself. Scientific American spoke to Cairo about the way harmonic analysis is like dropping stones into a still pond, her transgender identity and the reasons she loves mathematics. [An edited transcript of the interview follows.] Beyond 'points, lines, and waves,' how would you explain your field of study, harmonic analysis? Imagine that you're at a pond, and it's a very still pond, and you drop a stone into it. You see these circular waves spreading out. If you drop two stones in the pond, then you might notice this pattern called an interference pattern: instead of looking like circles, they overlap. You get high points, low points. And you get these interesting shapes [where they intersect]. What if you were to use a whole bunch of ripples—then what would you get? In harmonic analysis, you can actually prove that if you drop your stones in the right place in the pond, you can get any shape that you want. My specialty is known as Fourier restriction theory, which is the subdiscipline of harmonic analysis that I work in, where we ask what kind of objects can we build if we're only allowed to use certain kinds of waves. What if we're only allowed to drop the stones in certain parts of the pond? You won't be able to get just any object. In fact, you're only going to be able to get a relatively small family of objects. What the Mizohata-Takeuchi conjecture says is that the shape of the objects that we get are concentrated along lines. What does it mean to be 'concentrated along lines'? One way to think of the shape of objects is to ask: What is curvature? There are a few different ways you can define it. One possible way is to take a thin, long rectangle and ask how much of your circle can lie in this thin rectangle. What you'll find is that not very much of it can because it bends away, right? On the other hand, if you take something flat like the edge of a square, then you can get a whole side of that square just on one thin tube. So that means that the square is not as 'curved' as a circle. For the Mizohata-Takeuchi conjecture, we say, consider this object that we're building out of these waves. And we want to say that not very much is going to lie on shapes that do not contain very many lines or thin rectangles. So how did you manage to disprove this conjecture? I looked at these shapes, and one thing that I realized is that the specific kind of waves that are used are concentrated along thick rectangles. This is actually something that is well known. So you end up looking at these waves that are concentrated on rectangles: You take these waves, and they intersect each other, and they make these certain shapes, but here [instead of circle waves] we use rectangle waves. So then we have all of these rectangle waves meeting each other. What I realized is that the shape of where they meet is not quite at the right angle to agree with the direction that these rectangles are pointing in. And so this led me to a rather complicated construction using fractals to arrange these rectangles. The original fractal construction doesn't actually show up in your paper though. What was your final counterexample? What I found out is that if you arrange these waves by taking a high-dimensional hypercube and projecting it down into smaller-dimensional space and then taking only those waves that lie in your region, then this is how you can determine where to put them [to break the conjecture]. What first got you interested in math? I've always been interested in math. I think that, for me, mathematics is an art. In my childhood, I was somewhat lonely. Math was sort of there as a friend almost. I think that art cannot necessarily be a friend in every way that a friend can be, but I think art is like a friend. And so, for as long as I can remember, I've always loved mathematics. Tell me more about how math was a friend to you. I think a lot of people don't think of math as very friendly. There's an analogy that I like to make, which is to another form of art: painting. And I think that if one were to take a class on paint, you could memorize the dates and times at which various forms of paint were developed—and maybe even which paints were used by which painters. And then you can figure out what processes you can use to determine what type of paint it is. I imagine this is useful in art history, but this is not art.... I shouldn't say that. Maybe there is an art to learning about paint. I'm not going to claim that there isn't because I don't study paint. But I think that math is a little bit like that—in school, people learn about [the mathematical version of] paint; they're not learning about painting. Mathematics is comforting to me because it's a way of exploring—to explore ideas and to think about them and to build more ideas out of other ideas. What's comforting about that is that it's independent of the world in some ways. If I'm having a sad day, a happy day, if I move to Maryland (I did just move to Maryland), mathematics is still there, and it is still the same thing. It's also just something that can occupy my mind. You've mentioned to me that you're transgender. How has that affected your journey? I think that it's probably more relevant in my journey as a person than as a mathematician. Being trans has forced me to see things about the world that I maybe otherwise wouldn't have seen. It's made me see the world differently and made me see people differently and made me see myself differently. Fortunately, in the math community, I think that most mathematicians are fine with trans people. I think that it used to be more significant [in my day to day] than it is now. These days it doesn't really make much of a difference. Why have you decided to go on the record now as being trans? Trans visibility is important. People have ideas about who trans people are, and I think that it's best to broaden that. Maybe I'm also hoping that people who think that trans people are 'less' than cisgender people might find themselves questioning that. The other thing is that it's good for trans people to know that they're not alone. I think that part of what helps trans people realize that they're trans is to know that there are more options for who you can be as a trans person. That's important to me. Thank you so much for sharing that. Where is your favorite place to do math? If I'm trying to be productive in writing something down, then I like to be at my desk, and I like to listen to Bach. If I am just trying to think about ideas, then my favorite place to do that is somewhere where I don't have to pay attention to very much else. I could just be sitting down somewhere thinking about stuff, or I could be going for a walk outside. I also like to talk to other people about math, which is another kind of doing math. I really like to give presentations about mathematics. I have these handwritten slides with all these colors and drawings. Luckily, in harmonic analysis, I can give a presentation like this, and then everybody is so happy, and they tell me my slides are cute. What's next for your research? I'm working on a research project with my adviser on Mizohata-Takeuchi and adjacent stuff and about a sort of different thing: the local Mizohata-Takeuchi conjecture. The process of learning more about this kind of mathematics is pretty exciting—not just for me learning more about what's out there but for the math community as a whole to try to understand these kinds of things better. [That's] something that I'm excited about. Solve the daily Crossword
Scientific American
04-08-2025
- Science
- Scientific American
How Teen Mathematician Hannah Cairo Disproved a Major Mathematical Wave Conjecture
When Hannah Cairo was 17 years old, she disproved the Mizohata-Takeuchi conjecture, a long-standing guess in the field of harmonic analysis about how waves behave on curved surfaces. The conjecture was posed in the 1980s, and mathematicians had been trying to prove it ever since. If the Mizohata-Takeuchi conjecture turned out to be true, it would illuminate many other significant questions in the field. But after hitting wall after wall trying to prove it, Cairo managed to come up with a counterexample: a circumstance where the waves don't behave as predicted by the conjecture. Therefore, the conjecture can't be true. Cairo got hooked on the problem after being assigned a simpler version of the conjecture to prove as a homework assignment for a class she was taking at the University of California, Berkeley. 'It took me a while to convince [course instructor] Ruixiang Zhang that my proposal was actually correct,' she says. Now, under Zhang's advisement, she has a paper on the preprint server and was invited to present her results at the International Conference on Harmonic Analysis and Partial Differential Equations in El Escorial, Spain. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. Cairo says she loves talking about her research and giving presentations with colorful and descriptive slides (see examples below). When asked what she studies, Cairo says, in short, 'points, lines and waves.' Born and raised in the Bahamas, Cairo moved to California at the age of 16, where she began to take classes at U.C. Berkeley. Now, at 18 years old, she is on to a Ph.D. program at the University of Maryland to continue her research in Fourier restriction theory. Cairo has faced many difficulties in her journey, but she has found comfort and belonging in the field of mathematics and in the work itself. Scientific American spoke to Cairo about the way harmonic analysis is like dropping stones into a still pond, her transgender identity and the reasons she loves mathematics. [ An edited transcript of the interview follows. ] Beyond 'points, lines, and waves,' how would you explain your field of study, harmonic analysis? Imagine that you're at a pond, and it's a very still pond, and you drop a stone into it. You see these circular waves spreading out. If you drop two stones in the pond, then you might notice this pattern called an interference pattern: instead of looking like circles, they overlap. You get high points, low points. And you get these interesting shapes [where they intersect]. What if you were to use a whole bunch of ripples—then what would you get? In harmonic analysis, you can actually prove that if you drop your stones in the right place in the pond, you can get any shape that you want. My specialty is known as Fourier restriction theory, which is the subdiscipline of harmonic analysis that I work in, where we ask what kind of objects can we build if we're only allowed to use certain kinds of waves. What if we're only allowed to drop the stones in certain parts of the pond? You won't be able to get just any object. In fact, you're only going to be able to get a relatively small family of objects. What the Mizohata-Takeuchi conjecture says is that the shape of the objects that we get are concentrated along lines. What does it mean to be 'concentrated along lines'? One way to think of the shape of objects is to ask: What is curvature? There are a few different ways you can define it. One possible way is to take a thin, long rectangle and ask how much of your circle can lie in this thin rectangle. What you'll find is that not very much of it can because it bends away, right? On the other hand, if you take something flat like the edge of a square, then you can get a whole side of that square just on one thin tube. So that means that the square is not as 'curved' as a circle. For the Mizohata-Takeuchi conjecture, we say, consider this object that we're building out of these waves. And we want to say that not very much is going to lie on shapes that do not contain very many lines or thin rectangles. So how did you manage to disprove this conjecture? I looked at these shapes, and one thing that I realized is that the specific kind of waves that are used are concentrated along thick rectangles. This is actually something that is well known. So you end up looking at these waves that are concentrated on rectangles: You take these waves, and they intersect each other, and they make these certain shapes, but here [instead of circle waves] we use rectangle waves. So then we have all of these rectangle waves meeting each other. What I realized is that the shape of where they meet is not quite at the right angle to agree with the direction that these rectangles are pointing in. And so this led me to a rather complicated construction using fractals to arrange these rectangles. The original fractal construction doesn't actually show up in your paper though. What was your final counterexample? What I found out is that if you arrange these waves by taking a high-dimensional hypercube and projecting it down into smaller-dimensional space and then taking only those waves that lie in your region, then this is how you can determine where to put them [to break the conjecture]. What first got you interested in math? I've always been interested in math. I think that, for me, mathematics is an art. In my childhood, I was somewhat lonely. Math was sort of there as a friend almost. I think that art cannot necessarily be a friend in every way that a friend can be, but I think art is like a friend. And so, for as long as I can remember, I've always loved mathematics. Tell me more about how math was a friend to you. I think a lot of people don't think of math as very friendly. There's an analogy that I like to make, which is to another form of art: painting. And I think that if one were to take a class on paint, you could memorize the dates and times at which various forms of paint were developed—and maybe even which paints were used by which painters. And then you can figure out what processes you can use to determine what type of paint it is. I imagine this is useful in art history, but this is not art.... I shouldn't say that. Maybe there is an art to learning about paint. I'm not going to claim that there isn't because I don't study paint. But I think that math is a little bit like that—in school, people learn about [the mathematical version of] paint; they're not learning about painting. Mathematics is comforting to me because it's a way of exploring—to explore ideas and to think about them and to build more ideas out of other ideas. What's comforting about that is that it's independent of the world in some ways. If I'm having a sad day, a happy day, if I move to Maryland (I did just move to Maryland), mathematics is still there, and it is still the same thing. It's also just something that can occupy my mind. You've mentioned to me that you're transgender. How has that affected your journey? I think that it's probably more relevant in my journey as a person than as a mathematician. Being trans has forced me to see things about the world that I maybe otherwise wouldn't have seen. It's made me see the world differently and made me see people differently and made me see myself differently. Fortunately, in the math community, I think that most mathematicians are fine with trans people. I think that it used to be more significant [in my day to day] than it is now. These days it doesn't really make much of a difference. Why have you decided to go on the record now as being trans? Trans visibility is important. People have ideas about who trans people are, and I think that it's best to broaden that. Maybe I'm also hoping that people who think that trans people are 'less' than cisgender people might find themselves questioning that. The other thing is that it's good for trans people to know that they're not alone. I think that part of what helps trans people realize that they're trans is to know that there are more options for who you can be as a trans person. That's important to me. Thank you so much for sharing that. Where is your favorite place to do math? If I'm trying to be productive in writing something down, then I like to be at my desk, and I like to listen to Bach. If I am just trying to think about ideas, then my favorite place to do that is somewhere where I don't have to pay attention to very much else. I could just be sitting down somewhere thinking about stuff, or I could be going for a walk outside. I also like to talk to other people about math, which is another kind of doing math. I really like to give presentations about mathematics. I have these handwritten slides with all these colors and drawings. Luckily, in harmonic analysis, I can give a presentation like this, and then everybody is so happy, and they tell me my slides are cute. What's next for your research? I'm working on a research project with my adviser on Mizohata-Takeuchi and adjacent stuff and about a sort of different thing: the local Mizohata-Takeuchi conjecture. The process of learning more about this kind of mathematics is pretty exciting—not just for me learning more about what's out there but for the math community as a whole to try to understand these kinds of things better. [That's] something that I'm excited about.
The Star
03-08-2025
- General
- The Star
Contradictheory: 'Mathmo' for life – how math can be found everywhere
The other day a friend asked me, if I had to do it all over again, would I still choose to study maths at university? In some ways, I didn't really have much of a choice. Maths was the subject I earned a scholarship for, and the only one in which I got top grades across all my exams. I could have applied for an adjacent subject like engineering, or computer science, or even accountancy, but that would be like using a canon in a sword fight. It can work, but it's inelegant. The truth is that maths has been my weapon of choice in school for a very long time. I even learn music by looking at it from a mathematical perspective, finding patterns hidden within notes and chords, especially in Bach's work. For many years, it was also a bit of an obsession for me. It was what I would think of almost as soon as I woke up, as well as in those twilight moments as I was trying and failing to drop off to sleep. Eventually I did maths in university and was and became a 'mathmo' (Cambridge University slang for mathematicians). Law and history undergrads looked on with envy from their foot-long reading lists at us mathematicians whose weekly assignments rarely covered more than a couple of A4 sheets. We could discuss homework while calculating angles at the pool table or think about imaginary numbers while sprawled on the grass in the quad. Those heady days of full-time maths (at least in my head) are now behind me. I simply wasn't good enough to do it full time. Now I browse the odd YouTube video about the Langlands programme (about connections between various areas of mathematics), or solve cute puzzles about liars in green hats choosing doors, and these have become moments in my life rather than the mainstream. Occasionally, I do pay attention to news about mathematicians. For example, Hannah Cairo recently casually disproved a 50-year-old conjecture about the kinds of shapes you get when you combine a lot of very tiny waves. She presented her work in a deceptively whimsical-looking presentation and is due to start on her PhD later this year. She is also currently 17 years old. Another is Kate Wenqi Zhu who won a prestigious prize in numerical analysis, shutting up her haters who thought she was too trendy as a social media influencer to also be fluent in non-convex optimisation and tensors. But aside from these attention-grabbing headlines, what really caught my attention was how they approached maths. Were they more dedicated, more focused at maths than I, which is why they succeeded while I stagnated? Before she turned 17, Hannah was reading university-level textbooks for fun and emailed professors asking to join their classes. During the summer of the Covid-19 pandemic, she joined the Berkeley Math Circle online, and came to the realisation that doing maths was more like 'painting a picture' rather than memorising facts. It was a way to understand things, ask questions, and make friends. According to her, she uses maths to 'help other people, to make them happy'. Meanwhile, Kate was accepted into university at 15, and after graduating, she worked at JPMorgan and then Goldman Sachs. Life was good. She travelled, she ate at nice restaurants, she posted fashion shots online. But when she mentioned on Weibo she was also an Oxford graduate, the post got 1.2 billion views, along with a flood of accusations claiming she was faking it. In this dark period she turned back to maths. Solving problems brought her a sense of peace. This realisation prompted her to resign from her high-paying, high-flying job and return to Oxford to pursue her master's and doctorate degrees. Winning the prize to shut up the critics was probably just a nice bonus. I think these two stories give some insight about what it means to love mathematics. It's not about locking yourself in a room with a large blackboard to unlock the hidden secrets of the universe (or at least, not just about that). It's about finding the right place to share what you get from it with others. Indeed, I say this with the benefit of my own hindsight. While doing maths was fun, thinking about it 24/7 can have consequences. It's one thing lulling yourself to sleep with algebraic equations; it's another to be kept awake to the point of insomnia. But eventually, life coalesces, and I discovered that I also enjoyed teaching others and helping them understand the world around us. I ended up doing things like training people on how to use software, and helping kids prepare for tests – and writing columns about strange and wonderful topics, including maths. Maths is still very much with me, but it's part of a larger picture. It's one pillar among many others that supports the whole house. There is a very human aspect of being thrilled about something, falling in love with it, and then as life unfolds, finding the right places and times so it strengthens rather than consumes you. When this passion helps you better connect with the world at large, rather than isolate and exclude you, then it most definitely becomes something to embrace truly and tightly, even if it's just once in a while. Logic is the antithesis of emotion but mathematician-turned-scriptwriter Dzof Azmi's theory is that people need both to make sense of life's vagaries and contradictions. Write to Dzof at lifestyle@ The views expressed here are entirely the writer's own.
