Latest news with #mathematicians
Yahoo
2 days ago
- 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
WIRED
3 days ago
- Science
- WIRED
This New Pyramid-Like Shape Always Lands With the Same Side Up
Aug 10, 2025 7:00 AM A tetrahedron is the simplest platonic solid. Mathematicians have now made one that's stable only on one side, confirming a decades-old conjecture. The original version of this story appeared in Quanta Magazine. In 360 BC, Plato envisioned the cosmos as an arrangement of five geometric shapes: flat-sided solids called polyhedra. These immediately became important objects of mathematical study. So it might be surprising that, millennia later, mysteries still surround even the simplest shape in Plato's polyhedral universe: the tetrahedron, which has just four triangular faces. One major open problem, for instance, asks how densely you can pack 'regular' tetrahedra, which have identical faces. Another asks which kinds of tetrahedra can be sliced into pieces that can then be reassembled to form a cube. The great mathematician John Conway was interested not only in how tetrahedra can be arranged or rearranged, but also in how they balance. In 1966, he and the mathematician Richard Guy asked whether it was possible to construct a tetrahedron made of a uniform material—with its weight evenly distributed—that can only sit on one of its faces. If you were to place such a 'monostable' shape on any of its other faces, it would always flip to its stable side. A few years later, the duo answered their own question, showing that this uniform monostable tetrahedron wasn't possible. But what if you were allowed to distribute its weight unevenly? At first, it might seem obvious that this should work. 'After all, this is how roly-poly toys work: Just put a heavy weight in the bottom,' said Dávid Papp of North Carolina State University. But 'this only works with shapes that are smooth or round or both.' When it comes to polyhedra, with their sharp edges and flat faces, it's not clear how to design something that will always flip to the same side. Gábor Domokos discovers and builds new shapes to understand the world around us. Photograph: Ákos Stiller Conway, for his part, thought that such tetrahedra should exist, as some mathematicians recall him saying. But he ended up focusing on the balancing acts of higher-dimensional, uniformly weighted tetrahedra. If he ever wrote up a proof of his off-the-cuff 3D conjecture, he never published it. And so for decades, mathematicians didn't really think about the problem. Then along came Gábor Domokos, a mathematician at the Budapest University of Technology and Economics who had long been preoccupied with balancing problems. In 2006, he and one of his colleagues discovered a shape called the gömböc, which has the unusual property of being 'mono-monostatic'—it balances on just two points (one stable, the other unstable, like the side of a coin), and no others. Try to balance it anywhere else, and it will roll over to stand on its stable point. But like a roly-poly, the gömböc is round in places. Domokos wanted to know if a pointy polyhedron could have a similar property. And so Conway's conjecture intrigued him. 'How was it possible that there was an utterly simple statement about an utterly simple object, and yet the answer was far from immediate?' he said. 'I knew that this was very likely a treasure.' In 2023, Domokos—along with his graduate students Gergő Almádi and Krisztina Regős, and Robert Dawson of Saint Mary's University in Canada—proved that it is indeed possible to distribute a tetrahedron's weight so that it will sit on just one face. At least in theory. But Almádi, Dawson, and Domokos wanted to build the thing, a task that turned out to be far more challenging than they expected. Now, in a preprint posted online yesterday, they have presented the first working physical model of the shape. The tetrahedron, which weighs 120 grams and measures 50 centimeters along its longest side, is made of lightweight carbon fiber and dense tungsten carbide. To work, it had to be engineered to a level of precision within one-tenth of a gram and one-tenth of a millimeter. But the final construction always flip-flops onto one face, exactly as it should. The work demonstrates the important role of experimentation and play in research mathematics. It also has potential practical applications, such as in the design of self-righting spacecraft. 'I didn't expect more work to come out on tetrahedra,' Papp said. And yet, he added, the team's research allows mathematicians to 'really appreciate how much we didn't know and how thorough our understanding is now.' Tipping Point In 2022, Almádi, then an undergraduate aspiring to become an architect, enrolled in Domokos' mechanics course. He didn't say much, but Domokos saw in him a hard worker who was constantly in deep thought. At the end of the semester, Domokos asked him to concoct a simple algorithm to explore how tetrahedra balance. When Conway originally posed his problem, his only option would have been to use pencil and paper to prove, through abstract mathematical reasoning, that monostable tetrahedra exist. It would have been almost prohibitively difficult to pinpoint a concrete example. But Almádi, working decades later, had computers. He could do a brute-force search through a huge number of possible shapes. Eventually, Almádi's program found the coordinates for the four vertices of a tetrahedron that, when assigned certain weight distributions, could be made monostable. Conway was right. Krisztina Regős helped discover new properties of tetrahedra. Photograph: Courtesy of Krisztina Regős; Ms. Tara Inman Robert Dawson helped discover new properties of tetrahedra. Almádi found one monostable tetrahedron, but presumably there were others. What properties did they share? While that might seem like a simple question, 'a statement like 'A tetrahedron is monostable' cannot be easily described with a simple formula or a small set of equations,' Papp said. The team realized that in any monostable tetrahedron, three consecutive edges (where pairs of faces meet) would need to form obtuse angles—ones that measure over 90 degrees. That would ensure that one face would hang over another, allowing it to tip over. The mathematicians then showed that any tetrahedron with this feature can be made monostable if its center of mass is positioned within one of four 'loading zones'—much smaller tetrahedral regions within the original shape. So long as the center of mass falls inside a loading zone, the tetrahedron will balance on only one face. The gömböc, discovered in 2006, can stand on only two points, one stable, the other unstable. Mathematicians have continued to search for other shapes with intriguing balancing properties. Photograph: Gábor Domokos Achieving the right balance between the weight of the loading zone and the weight of the rest of the tetrahedron is easy in the abstract realm of mathematics—you can define the weight distribution without a care for whether it's physically possible. You might, for instance, let parts of the shape weigh nothing at all, while concentrating a large amount of mass in other parts. But that wasn't entirely satisfying to the mathematicians. Almádi, Dawson, and Domokos wanted to hold the shape in their hands. Was it possible to make a monostable tetrahedron in the real world, with real materials? Getting Real The team returned to their computer search. They considered the various ways in which monostable tetrahedra might tip onto their stable face. For instance, one kind of tetrahedron might follow a very simple path: Face A tips to Face B, which tips to Face C, which tips to Face D. But in a different tetrahedron, Face A might tip to Face B, and both Face B and Face D will tip to Face C. The loading zones for these different tetrahedra look very different. The team calculated that to get one of these 'falling patterns' to work, they would need to construct part of the shape out of a material about 1.5 times as dense as the sun's core. While studying to be an architect, Gergő Almádi was drawn to a decades-old geometry problem. Photograph: Réka Dolina They focused on a more feasible falling pattern. Even so, part of their tetrahedron would have to be about 5,000 times as dense as the rest of it. And the materials had to be stiff—light, flimsy materials that could bend would ruin the project, since it's easy to make a round or smooth shape (like the roly-poly) monostable. In the end, they designed a tetrahedron that was mostly hollow. It consisted of a lightweight carbon fiber frame and one small portion constructed out of tungsten carbide, which is denser than lead. For the lighter portions to have as little weight as possible, even the carbon fiber frames had to be hollow. With this blueprint in hand, Domokos got in touch with a precision engineering company in Hungary to help build the tetrahedron. They had to be incredibly accurate in their measurements, even when it came to the weight of the tiny amounts of glue used to connect each of the shape's faces. Several frustrating months and several thousand euros later, the team had a lovely model that didn't work at all. Then Domokos and the chief engineer of the model spotted a glob of stray glue clinging to one of its vertices. They asked a technician to remove it. About 20 minutes later, the glue was gone and Almádi received a text from Domokos. 'It works,' the message read. Almádi, who was on a walk, started jumping up and down in the street. 'Seeing the lines on the computer is very far from reality,' he said. 'That we designed it, and it works, it's kind of fantastic.' 'I wanted to be an architect,' he added. 'So this is still very strange for me—how did I end up here?' In the end, the work on monostable tetrahedra didn't involve any particularly sophisticated math, according to Richard Schwartz of Brown University. But, he said, it's important to ask this kind of question in the first place. It's the kind of problem that's often easiest to overlook. 'It's a surprising thing, a leap, to conjecture that these things would exist,' Schwartz said. At the moment, it's not clear what new theoretical insights the model of the monostable tetrahedron will provide—but experimenting with it might help mathematicians uncover other intriguing questions to ask about polyhedra. In the meantime, Domokos and Almádi are working to apply what they learned from their construction to help engineers design lunar landers that can turn themselves right side up after falling over. In any case, sometimes you just need to see something to believe it, Schwartz said. 'Even for theoretical math, geometry especially, people are kind of right to be skeptical because it's quite hard to reason spatially. And you can make mistakes, people do.' 'Conway didn't say anything about it, he just suggested it—never proved it, never proved it wrong, nothing. And now here we are, I don't know, 60 years later,' Almádi said. 'If he were still alive, we could put this on his desk and show him: You were right.' Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
Wall Street Journal
28-07-2025
- Science
- Wall Street Journal
The High-Schoolers Who Just Beat the World's Smartest AI Models
The smartest AI models ever made just went to the most prestigious competition for young mathematicians and managed to achieve the kind of breakthrough that once seemed miraculous. They still got beat by the world's brightest teenagers.
Daily Mail
13-07-2025
- Science
- Daily Mail
Easy-looking math sum leaves people confused - can you solve it without a calculator?
Every so often it's good to exercise your brain with a math problem or two that forces you to recall principles you learned decades prior. As elementary as it might feel, you'd be surprised how easy it is to be stumped by a seemingly simple equation. This problem, posted on X, in particular may be challenging for even the best of mathematicians. Can you solve it without a calculator? 5+5 x 5+5 At first it may look easy, but it's important to go back to basics before diving in head first. The acronym PEMDAS can be handy when solving a sum like this one. Its letters spell out the order in which you should solve complex equations. The 'p' stands for parenthesis, then exponents, multiplication and division, and then adding or subtracting. Now, give it a try for yourself. Using the rules of PEMDAS, the first step is to solve the sum at the center of the problem, no matter how unnatural it may feel: 5 x 5 = 25 Now, the problem reads: 5+25+5 With simple addition from left to right, the final sum is easy: 5+25=30 30+5= 35 How did you do? If you couldn't solve it, don't worry, there are a few common mistakes that could have led to a different answer. The first, is adding on both sides before multiplying in the middle. Adding first on the left: 5+5=10 Making the problem: 10x5+5 Then, adding on the right: 5+5=10 That makes the solution: 10x10=100 Unfortunately, that method totally ignores the rules of PEMDAS, rendering the answer incorrect. Another easy error would be to solve from left to right, which may feel natural at first. 5+5=10 That leaves: 10x5+5 Then: 10x5=50 Which would make the solution: 50+5=55 But PEMDAS is a fool-proof way to guarantee the right answer every time and tease your brain with all-too-familiar math tricks.
New York Times
11-07-2025
- General
- New York Times
Test Your Math Knowledge
'Math, Revealed,' our four-part series exploring the mathematics behind everyday objects and experiences, recently came to a smashing conclusion. One installment took a spin through 'taxicab geometry,' a wacky but vital corner of mathematics in which circles aren't round and pi equals 4. Another journey began with apples and pentagrams and led to Leonardo da Vinci, the golden ratio and ideal positioning of belly buttons. We had great fun on this adventure and hope to resume it before too long. In the meantime, here's a quiz to test what you learned and your general math knowledge. Enjoy! 1. In taxicab geometry, circles don't look round — they form sharp, angular shapes. What shape do they resemble? A triangle A diamond A hexagon A star An octagon 2. In taxicab geometry, even the value of pi is a surprise. What is it? About 3.14 About 1.41 Exactly 2 Exactly 3 Exactly 4 3. Mathematicians have long been fascinated by a special number that describes self-similar proportions. What is the approximate value of this 'golden ratio'? 1.41 1.50 1.62 1.75 2.00 4. Leonardo da Vinci's 'Vitruvian Man' has been analyzed endlessly for hidden patterns. According to a 2015 study, does the navel divide the figure according to the golden ratio? Yes, almost exactly. Yes, but only approximately. No, it follows a 2:1 ratio. No, it follows a 3:2 ratio. No, but the golden ratio is mentioned in Leonardo's notes. 5. In the densest possible arrangement of soda cans standing on a flat surface, each can in the middle touches the same number of neighbors. How many is it? Three Four Five Six Eight 6. In 2022, a mathematician won a Fields Medal for solving a problem about how to pack spheres tightly in eight dimensions. Who did it? Terence Tao Maryna Viazovska Thomas Hales Maryam Mirzakhani Ingrid Daubechies 7. The first four triangular numbers are 1, 3, 6 and 10. Why is 10 considered a triangular number? It's divisible by 3. It's the sum of three primes. It appears in the Pythagorean theorem. You can arrange 10 dots in an equilateral triangle. It's shaped like a triangle on the number line. 8. In 1672, Gottfried Wilhelm Leibniz found a clever way to add the reciprocals of all the triangular numbers. What sum did he get? 1 + 1/3 + 1/6 + 1/10 + ... 2 3 e Pi Infinity 9. At a wedding reception, the bride seats eight of her ex-boyfriends together at a table. (This actually happened to me once.) If each ex-boyfriend shakes hands with each of the others, how many handshakes occur in total? 28 32 36 40 56 10. Four bugs start in the corners of a square that measures 1 foot by 1 foot. Each bug chases its clockwise neighbor, always crawling directly toward the neighbor's current position. If they all crawl at the same speed, how far does each bug travel before they all meet at the center? Not enough information is given. I never took calculus. Help! 1 foot 2 feet 1.41 feet Questions and answers 1 through 8: photo illustrations by Jens Mortensen for The New York Times; answer 6: photo by Brendan Hoffman for The New York Times; question 9: photo illustration by The New York Times, source photo via Alamy; question 10: photo illustration by The New York Times, source photos by Balarama Heller for The New York Times. Produced by Alan Burdick, Alice Fang, Marcelle Hopkins and Matt McCann.
