Mathematicians Solve Multidimensional Shape-Slicing Dilemma
Bourgain's slicing problem asks whether every convex shape in n dimensions has a 'slice' such that the cross section is bigger than some fixed value. For three-dimensional objects, this is like asking whether an avocado of a given size, no matter the exact shape, can always be split into two halves with each side revealing at least some sizable slice. Bourgain, a titan of mathematics, is said to have spent more time on this problem than any other; although it may seem deceptively easy to resolve in the physical world's two or three dimensions, it quickly balloons in difficulty when we consider four or five. This added complexity makes determining anything in n-dimensional space seem impossible. 'If you believe in this so-called curse of dimensionality, you might just give up,' Klartag says. Fortunately, he adds, he and Lehec 'belong to a different school of thought.'
The pair's breakthrough builds on recent progress by mathematician Qingyang Guan of the Chinese Academy of Sciences, who approached the problem with a technique based on physics rather than geometry. Specifically, Guan showed that modeling how heat diffuses out of a convex shape can reveal hidden geometric structures. Researchers could calculate filling any convex shape with warm gas and carefully observe the heat's dissipation according to physical laws. Guan's key insight—a precise limit on how rapidly the rate of dissipation changes during this heating process—proved to be just what Klartag and Lehec needed. 'Guan's bound tied together all the other key facts' known for the problem, says mathematician Beatrice-Helen Vritsiou of the University of Alberta.
[Sign up for Today in Science, a free daily newsletter]
The result let Klartag and Lehec resolve the problem in only a few days. Klartag notes that 'it was lucky because we knew [Guan's result] was exactly one of the things we needed' to connect several seemingly disparate approaches to the puzzle. With this final piece in place, the geometry of convex bodies in high dimensions is now a little less mysterious—although, as always in mathematics, each new slice reveals more questions to explore.
Hashtags
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
Yahoo
2 days ago
- Yahoo
1906 shipwreck is filled with rare oysters to boost marine species
A cargo ship lying at the bottom of the sea off the Belgian coast has been fitted with a new treasure chest: a stash of rare flat oysters. Molluscs have mostly disappeared from the North Sea due to human activity, including overfishing. Now, a Belgian project is trying to reintroduce it in a move scientists believe will help boost other marine species. "We have to bring them back because they are essential elements in our marine ecosystems," Vicky Stratigaki, an engineer working on the restoration project, told AFP. In mid-July, a load of 200,000 oyster larvae attached to biodegradable materials was deposited about 100 feet under the sea in the ship's hull. The environmental project, named Belreefs, aims to turn the wreckage into a biodiversity sanctuary. Flat oysters form reefs that purify water and that other sea animals, from fish to algae, use as breeding and feeding grounds, explained Stratigaki. "There is a lot of predation in the sea, it's a wild environment," she said, with about 30,000 of the oyster larvae expected to survive their first year at sea. "Then they will start reproducing, extending the reef and also supporting the biodiversity of the reef." The laying of the oyster stash is the culmination of two years' work for the Belgian government project, which is supported by European Union funding. "Until around the 1850s, the North Sea and the European waters were full of these oyster reefs," Stratigaki explained. Then overfishing, the spreading of an imported parasite called Bonamia and "climatic adverse effects" caused them to disappear, she said. The 1906 wreckage, located about 20 miles off the coastal city of Ostend, was selected to house the pilot as fishing and other disruptive activities are banned around it. "In Belgium every wreck that is for more than one hundred years on the sea bottom gets protected automatically as cultural heritage, because it's nice for divers to go there," said Merel Oeyen, a marine environment expert at the Belgian ministry of health. "It's also a hot spot for biodiversity." A 2023 paper published in BioScience found that shipwrecks provide important ecological resources for a wide variety of organisms, from tiny microbes to large marine creatures. "Small fish and mobile crustaceans often find shelter in the crevices of the sunken material, and larger baitfish and predators use shipwrecks as feeding grounds and rest stops as they swim from one place to another," according to NOAA, which helped conduct the study. However, scientists also warned shipwreck can also cause damage to existing marine life in the area, or carry harmful cargo, such as oil. Still, the study's author, Avery Paxton, said shipwrecks can have "second lives" as homes to a variety of marine life. "A ship's transformation from an in-service vessel into a thriving metropolis for marine life has a fairy-tale quality to it," Paxton said in an article published by the Washington Post after the study was released. What shocked "Matlock" star Kathy Bates? A new you: The science of redesigning your personality "Somebody Somewhere" star Bridget Everett Solve the daily Crossword
Yahoo
4 days ago
- Yahoo
Can U.S. Math Research Survive NSF Funding Cuts?
A 72 percent reduction in federal funding is devastating to math research. The American Mathematical Society is offering $1 million in backstop grants—but it's likely not enough. Mathematics research typically requires few materials. To explore the secrets of prime numbers, investigate unimaginable shapes or elucidate other fundamental mysteries of our universe, mathematicians don't usually need special labs and equipment or to pay participants in clinical trials. Instead funding for mathematicians goes toward meetings of the mind—conferences, workshops and institutes where they gather for intensive sessions to work out math's knottiest problems. Funding also supports the stipends of research fellows, postdoctoral scholars and promising early-career mathematicians. But under the Trump administration's National Science Foundation, much of this funding is being revoked or cut—which, according to experts, could be catastrophic for the present and future of the field. In one recent example, the NSF canceled funding for the Association for Women in Mathematics' research symposium in Wisconsin just four business days before the event was set to begin in May. The threat to this event catalyzed the American Mathematical Society to offer $1 million in backstop grants to support programs whose federal funding has been cut or remains in limbo. These grants are meant to provide a financial safety net that will temporarily allow math programs, researchers and departments to continue operating—but it's not a permanent solution. (Disclosure: The author of this article currently has a AAAS Mass Media Fellowship at Scientific American that is sponsored by the American Mathematical Society.) 'The funding cut is severe, and all of mathematics will be impacted,' says Raegan Higgins, president of the Association for Women in Mathematics and a mathematician at Texas Tech University. [Sign up for Today in Science, a free daily newsletter] Movies and television shows often portray mathematicians scribbling on chalkboards in seclusion, but that picture is often far from accurate. 'None of us work in isolation,' Higgins says. In fact, mathematicians rely heavily on their ability to gather and discuss ideas with their peers—perhaps even more than researchers in other fields do. For mathematicians, conferences, workshops and research talks are not just opportunities to share research and network but also crucial moments to work out tough problems together with colleagues, pose field-propelling questions and generate new ideas. 'It's a thinking science, [and] it's a communication science, so we rely on being together to share ideas and to move the needle forward,' says Darla Kremer, executive director of the Association for Women in Mathematics. According to John Meier, CEO of the American Mathematical Society, 'the ability of mathematicians to gather and talk with each other is absolutely central to the vitality of the field.' Federal dollars, largely through the NSF, are responsible for a significant portion of math funding. But a lot of that funding is disappearing under the Trump administration. In April NSF staff members were instructed to 'stop awarding all funding actions until further notice.' Over the past 10 years, on average, the NSF has awarded $113 million in grants to mathematics by May 21 of each year. This year the NSF has awarded only $32 million, representing a 72 percent reduction. By this metric, mathematics is one of the most deeply affected subjects, second only to physics, which has seen an 85 percent reduction. The administration is also canceling and freezing funding that it had previously promised to researchers. More than $14 million of funding already promised to mathematics programs was revoked earlier this year, according to an analysis by Scientific American. In response to a request for comment, the National Science Foundation told Scientific American that 'the agency has determined that termination of certain awards is necessary because they are not in alignment with current NSF priorities and/or programmatic goals.' This withdrawal of grants is eroding trust and seeding uncertainty, experts say, and it comes with long-term consequences. Even if funding gets renewed again later, it can be very difficult for halted programs to recover. 'If you have to shut down a lab and mothball it, that actually takes time and effort,' Meier says. 'You can't just walk in two weeks later, flip a switch and have everything running again. You've got to rebuild it.' Even in mathematics, that process of rebuilding is time-intensive and not always possible if the space has been reallocated or the people have moved on. American Mathematical Society leadership fears these cuts will hurt young mathematicians the most. Like in the sciences, the funding cuts are eliminating research experiences and supportive programming for undergraduates, fellowships for graduate students and positions for postdoctoral researchers. Travel funding for conferences is also disappearing, which leaves young researchers to choose between shelling out for airfare and lodging they can't really afford and forgoing major career and research building opportunities. As these opportunities disappear, young mathematicians are beginning to look elsewhere—either to more lucrative jobs in the private sector or to more supportive countries. 'We worry about diminishing opportunities in the United States and people early in their career deciding that maybe there's a more profitable venue for them to pursue mathematics in another country,' Meier says. 'We love good mathematics wherever it arises, but we'd really like to see a lot of it arising in the United States. We think that's very, very important.' The $1 million in backstop grants can't fill the hole left by the more than $14 million in promised funding that has been denied or the more than $80 million in reduced funding so far this year. But it might be enough to keep many projects afloat simply by offering guaranteed access to funds in a turbulent time. 'I think one of the great difficulties that we're dealing with right now is the high level of uncertainty,' Meier says. Some mathematicians, for example, simply don't know whether their projects are still being funded or not. In some applications for the backstop grants, researchers 'basically talk about being ghosted,' Meier explains. 'They say, 'I can't actually verify that we no longer have funding. I can only tell you my program officer [at the NSF] isn't replying to my request for information.'' Meier hopes the grants can provide some backup for programs that aren't sure where they stand with the NSF. Without it, researchers, universities and independent organizations may find themselves facing impossible situations. Do they pay their research assistants, run their conferences and continue to fund travel out of pocket, assuming all the financial risk themselves and hoping the grants come through? Or do they halt their projects, losing valuable momentum and perhaps leaving important stakeholders unpaid for their work? Still, the backstop grants are a one-time offering—not a sustainable source of funding for an imperiled field. 'I really view them as trying to take a little bit of the sharp edges off of the sudden loss of funding, as opposed to anything that could sustain the field long-term,' Meier explains. The effects of the Trump administration's cuts to mathematics research—unlike research on, say, Alzheimer's disease, vaccines or climate change—may not be the most immediately concerning to human health and safety. But experts like Meier say that ignoring the role mathematics plays in that development is shortsighted. As a spokesperson of the NSF itself put it in response to an inquiry about the organization's changing priorities (and as the agency has said on its website), 'Mathematical sciences are crucial to everyday society and play an essential role in the innovation engine that drives the U.S. economy, strengthens national security and enhances quality of life.' And the search for the answers to math's biggest mysteries also seeds development in physics, earth science, biology, technology, and more. Any progress we make on these questions in the future, Meier says, is 'based entirely [on what] we are doing in research mathematics right now.' Solve the daily Crossword
Yahoo
5 days ago
- Yahoo
Can You Drink Saturn's Rings?
It's certainly possible to consume water sourced from the icy rings of Saturn, but doing so safely may require extra steps In November 2024 I was interviewed for a marvelous NPR podcast called Living On Earth about my latest popular science book, Under Alien Skies. While prepping for the show, one of the producers asked me a question that was so deceptively simple, so wonderfully succinct, and came from such an odd direction that I was immediately enamored with it. Can you drink Saturn's rings? After pausing for a moment to savor the question, I replied with one of my favorite responses as a scientist and science communicator: 'I don't know. But I'll try to find out.' [Sign up for Today in Science, a free daily newsletter] So I did. And to my delight, the nuanced answer I found is another personal favorite: Yes! But no. Kinda. It depends. I love this sort of answer because it arises when the science behind a seemingly easy question is very much not so simple. So please grab a frosty glass of (locally sourced) ice water, sit back and let me explain. Saturn's rings were likely first seen by Galileo in 1610. His telescope was fairly low-quality compared with modern equipment. And through its optics, all he could see were a pair of blobs, one on each side of the planet's visible face; he referred to them as Saturn's 'ears.' It wasn't until a few decades later that astronomers realized these 'ears' were actually a planet-encircling ring. Much was still unclear, but one thing was certain: the ring couldn't be solid. The speed at which an object orbits a planet depends on its distance from that world, and Saturn's ring was so wide that the inner edge would orbit much more rapidly than its outer edge, which would shear anything solid apart. Astronomers came up with a variety of different ideas for the structure, including a series of solid ringlets or even a liquid. It wasn't until the mid-1800s that the great Scottish physicist James Clerk Maxwell proved none of these would be stable and instead proposed what we now know to be true: the structure around Saturn was made of countless small particles, which were far too tiny to be seen individually from Earth. Further, these small objects form not just one ring but several, and these major rings are designated by letters in order of their discovery. The A ring is the outermost bright ring. Just interior to it is the bright and broad B ring, which contains most of the entire ring system's mass. Interior to that is the darker C ring, which leads down to the faint D ring that extends almost to the upper atmosphere of Saturn itself. In total these rings stretch across nearly 275,000 kilometers—two thirds of the Earth-moon distance! Despite their immense sprawl, the rings are almost impossibly flat, in many places just about 10 meters thick. Seen exactly edge-on, they look like a narrow line cutting across the planet. But what are they made of? Observations over the centuries have revealed that the main constituent of the rings is startlingly simple: water ice! Good ol' frozen H2O is extremely common in the outer solar system and makes up most of many moons and other small bodies there. In fact, in situ observations performed by the Cassini spacecraft—which orbited Saturn for more than a dozen years—showed that in some places the rings were made of almost perfectly pure water ice. Even better, most of the ring bits are a few centimeters across or smaller—the size of ice cubes, so they're already conveniently packaged. Sounds great! All you need to do then is scoop up some chunks, warm them—a lot (the average temperature of the rings is about –190 degrees Celsius)—and have yourself a nice, refreshing sip. But not so fast. This is where it gets more complicated. The spectra of the rings also show that they aren't made of absolutely pure ice. There's other material in the rings, and even though we're typically talking about contamination of less than 1 percent by mass, it's not clear what this stuff is. Scientists' best guess is that it comes from the impacts of micrometeorites, tiny particles whizzing around the outer solar system. This material is therefore likely composed of silicates (that is, rocks) or abundant metals, namely iron. Neither of these will harm you, although the U.S. Environmental Protection Agency recommends no more than 0.3 milligram of iron per liter of potable water (to avoid a metallic taste). You'd better run a magnet over your ring water before you drink it—and you should probably filter out any silicate sediments while you're at it. On the other hand, the rings' spectra suggest the presence of some unknown carbon-based contaminants as well. One likely candidate would be complex organic molecules called polycyclic aromatic compounds, or PAHs, which are relatively prevalent in space; many giant stars blow out PAH-laced winds as they die. One molecule that is commonly present in PAHs is cyanonaphthalene, which is considered carcinogenic. (It's unclear, though, how much exposure poses risks to humans—or, for that matter, whether this specific molecule actually exists in the rings.) It's best to be cautious and avoid these potential contaminants by picking your rings carefully. The abundance of water ice is highest in the outer A and middle B rings, for example, whereas the C and D rings appear to be the most contaminated. So, generally speaking, it'd probably be better to opt for ice from A or B while skipping C and D entirely. There could also be other ices in the rings, too, including frozen methane and carbon dioxide. Methane should bubble out when the ice is liquefied, and of course CO2 is what makes carbonated beverages fizzy. That might actually add a fun kick to drinking from the rings! There are other rings, too, outside the major ones we've already mentioned. For example, Saturn's icy moon Enceladus boasts dozens of geysers that blast liquid water from its interior out into space. This material forms a faint, fuzzy ring (the E ring) that, again, is mostly water ice but also contains small amounts of silicates—and noxious ammonia—so I wouldn't recommend it. Still, all in all, it looks like—if carefully curated and cleaned—Saturn's rings are indeed drinkable! How much water is there in the rings, then? The total mass of the rings is about 1.5 × 1019 kilograms, which, correcting for the density of ice and the removal of contaminants, should yield about 10 quintillion liters of water—enough to keep every human on Earth well hydrated for more than a million years. Eventually, if and when humans start to ply the interplanetary space-lanes, they'll need extraterrestrial sources of water because lifting it from Earth is difficult and expensive. Saturn's rings might someday become a popular rest stop. And, oh my, what a view visitors would have as they filled up! My thanks to my friend and outer solar system giant planet astronomer Heidi Hammel for her help with this article and to El Wilson for asking me this terrific question! Solve the daily Crossword
