How Teen Mathematician Hannah Cairo Disproved a Major Conjecture in Harmonic Analysis
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 arXiv.org 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
Hashtags
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
CBS News
2 hours ago
- CBS News
Warming water globally leads to changes off the coast of Massachusetts
By AARON PARSEGHIAN As ocean temperatures continue to rise, the water around Massachusetts is warming with it and scientists said the changes are becoming increasingly visible along our coast. Colleen Bowie, who has visited Revere Beach for decades, said she has noticed differences over the years. "It's still beautiful, we're lucky to be here," Bowie said. "I appreciate it, but I wish it wasn't changing so much." Researchers said this summer's ocean temperatures in the Bay State are part of a clear climate change trend. That warmth may be behind recent unusual sightings and health advisories — from a manatee spotted in Mashpee earlier this month to an alert in Falmouth about vibrio vulnificus, a bacteria that can cause a flesh-eating disease typically found along the Gulf Coast. Dr. Kristina Dahl, Vice President of Science at Climate Central, said warming waters are allowing new species to migrate north and creating conditions for harmful bacteria and algal blooms to thrive. "With future global warming, we'll continue to see the oceans warm," Dahl said. "And we'll continue to see this poleward march of many species and the sorts of things you're seeing with respect to bacteria, or harmful algal blooms offshore. I lived on Cape Cod for years and the idea that you would see a manatee is just nuts. Obviously, animals are unpredictable. So you get things like isolated one-off animals showing up in weird places just somewhat naturally. But given that manatees are a warm water species, you don't expect to see them in Massachusetts normally but it could be a sign or symptom of our warming temperatures." The warming trend is already driving traditional Massachusetts marine life, such as cod and lobster, further north in search of colder waters, according to researchers in Woods Hole. "One of the remarkable things is just how quickly the oceans are changing," said Glen Gawarkiewicz, senior scientist at the Woods Hole Oceanographic Institution. "It's absolutely important that we keep up our ocean observations." Over the last two decades, the frequency of warm salty water breaking off from the Gulf Stream and making its way up the east coast has increased 70%, according to Gawarkiewicz. "We're fortunate now to be working more closely with the commercial fishing industry. And we have more and more fishing vessels that are taking temperature and salinity profiles," he added. "But a lot of these changes have been unexpected."
Yahoo
3 hours ago
- Yahoo
Ultra-processed foods that fuel colon cancer — and healthy alternatives that may offset the damage
Growing evidence has linked the rise in colon cancer cases to ultra-processed foods that fill our diets. Researchers found tumors are fueled by Inflammatory compounds and a lack of compounds linked to healing. Foods such as leafy greens and fish may help pinpoint disease-fighting molecules to stave off cancer. As colon cancer diagnoses are on the rise in younger people, researchers are discovering that certain foods may help or hurt our odds of developing the disease. A team of scientists from the University of South Florida and Tampa General Hospital Cancer Institute may have found clues in compounds called bioactive lipids, molecules that can increase or decrease inflammation in the body. In a study published late last year, the researchers analyzed more than 100 tumor samples from patients and found that they contained a higher proportion of these inflammation-causing compounds compared with healthy tissue. Previous evidence has linked increased inflammation to diets high in ultra-processed food, such as chips, sausages, packaged desserts, and refined carbs. As processed food takes up a growing share of the American diet (more than half, recent CDC data shows), a growing number of younger people are being diagnosed with colon cancer. It's now the second-leading cause of cancer-related death in the US. But inflammation isn't just a red flag that the body is under stress: tracking it can help doctors and scientists find ways to bolster the immune system against disease, slowing or stopping tumor growth by cutting back on inflammatory compounds and boosting healthy ones. For instance, the research team found that the tumors were lacking in molecules associated with healing and lowering inflammation. One source of these molecules is our diet, which includes foods such as leafy greens and seafood rich in omega-3 fatty acids. Eating more of these gut-friendly foods and fewer packaged meals may help keep inflammation at bay for better health. "If the molecules are coming from processed food products, they directly imbalance the immune system and drive chronic inflammation," Ganesh Halade, a professor at the University of South Florida Health Heart Institute who coauthored the study, said in a press release. "Our bodies are designed to actively resolve inflammation through bioactive lipid compounds derived from the healthy fats, like avocados, that we consume." While people who follow a "clean" diet can still get cancer, understanding the role of food could give us more tools to fight cancer. Fighting cancer with fish oil Cancer is like a "chronic wound that won't heal," Dr. Timothy Yeatman, a professor of surgery at the University of South Florida who's the senior author of the tumor study, said in the release. A daily diet of ultra-processed foods can make it harder for the body to fight off tumors because of the increased inflammation, he said in a press release. In particular, foods that cancer doctors avoid include processed meats and sugary treats, both of which are linked to higher odds of illnesses, including cancer and cardiovascular disease. While it's not news that spinach and seafood are better for you than bacon and doughnuts, zeroing in on the link between food and inflammation unlocks more effective strategies to stave off disease in the future. For example, the Tampa General Hospital Cancer Institute has conducted early trials of a modified form of fish oil with promising results for reducing inflammation. "This has the potential to revolutionize cancer treatment, moving beyond drugs to harness natural healing processes," Yeatman said. "It's a vital step toward addressing chronic inflammation and preventing diseases before they start." For now, one of the best ways to prevent cancer is to catch it early with routine testing. As far as eating your way to better health, evidence suggests your best bet is sticking to mostly whole, unprocessed foods such as veggies, fruits, legumes, lean proteins, and whole grains for a longer, healthier life."This has the potential to revolutionize cancer treatment, moving beyond drugs to harness natural healing processes," Yeatman said. "It's a vital step toward addressing chronic inflammation and preventing diseases before they start." In the meantime, there's evidence that your best bet for a longer, healthier life is sticking to a diet of mostly whole, unprocessed foods such as veggies, fruits, legumes, lean proteins, and whole grains. Read the original article on Business Insider Solve the daily Crossword
Yahoo
4 hours ago
- Yahoo
Eating this surprising fruit might be the key to getting a better night's sleep
Eating one avocado a day might be the key to a better night's sleep. Adults who included the fruit -- and yes, it's a fruit -- in their daily diet over the course of just six months reported better quality sleep compared to people who consumed fewer than two avocados each month, researchers said. 'Sleep is emerging as a key lifestyle factor in heart health, and this study invites us to consider how nutrition — and foods like avocado — can play a role in improving it,' Dr. Kristina Petersen, an associate professor of nutritional sciences at Penn State University, explained. The recent findings on sleep were tied to avocado's beneficial nutrients including tryptophan, folate, and magnesium. An essential mineral, magnesium is involved in muscle contraction and relaxation. Tryptophan and folate are involved in producing melatonin, the hormone that regulates our sleep cycles. These conclusions were drawn following an assessment of 969 U.S. adults who had what is considered a larger waistline -- 35 inches or more in women and 40 inches or more in men. The study was supported by the Avocado Nutrition Center, although the agency had no role in data collection, analysis, or interpretation. The researchers had initially intended to focus on heart health alone, before participants reported sleep-related benefits. "This was a cardiovascular health trial, making the sleep benefits more credible since they emerged as unexpected secondary findings in a well-designed randomized controlled trial," Dr. John Saito, a spokesperson for the American Academy of Sleep Medicine and pulmonologist at Children's Hospital of Orange County, told Verywell this week. Eating avocados daily was also associated with a better diet and lower cholesterol levels, the researchers reported, building on similar research from 2022. Studies have previously found that disrupted sleep is linked to high levels of cholesterol: the waxy, fat-like substance that can lead to heart attack and stroke. Improved heart health can help people to get better sleep, and getting enough sleep can also improve heart health and slash the risk of disease, according to the Centers for Disease Control and Prevention. Avocado's healthy fat and fiber contributed to positive cardiovascular impacts. The healthy fat can also help to reduce cholesterol levels, and lower the risk for potentially life-threatening cardiac events. The fiber is tied to a healthy gut and reduced risk of death from any cause. Adults should consume at least 25 grams of fiber each day, according to Harvard Medical School. Of course, healthy fat is still fat, and avocados are packed with both nutrients and calories. An entire large avocado can add upward of 400 calories to your daily diet, the Cleveland Clinic says. But, the fruits are still considered to be healthy when consumed in moderation like being added to a morning smoothie or a salad at lunch. Half an avocado has more potassium than a banana, the Harvard T.H. Chan School of Public Health said. Humans need potassium to help fend off high blood pressure, which can result in kidney disease, eye damage, coronary artery disease, and other complications, if it's left untreated. While the findings of the new study cannot be generalized to all populations, Petersen noted that heart health is influenced by many factors, such as fitness and genetics. 'This is an encouraging step in expanding the science around avocados and the potential benefits of consumption,' she said.
