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One afternoon in January 2011, Hussein Mourtada leapt onto his desk and started dancing. He wasn’t alone: Some of the graduate students who shared his Paris office were there, too. But he didn’t care. The mathematician realized that he could finally confirm a sneaking suspicion he’d first had while writing his doctoral dissertation, which he’d finished a few months earlier. He’d been studying spec
On warm summer nights, green lacewings flutter around bright lanterns in backyards and at campsites. The insects, with their veil-like wings, are easily distracted from their natural preoccupation with sipping on flower nectar, avoiding predatory bats and reproducing. Small clutches of the eggs they lay hang from long stalks on the underside of leaves and sway like fairy lights in the wind. The da
The Neumayer III polar station sits near the edge of Antarctica’s unforgiving Ekström Ice Shelf. During the winter, when temperatures can plunge below minus 50 degrees Celsius and the winds can climb to more than 100 kilometers per hour, no one can come or go from the station. Its isolation is essential to the meteorological, atmospheric and geophysical science experiments conducted there by the m
More than a century after Ernest Rutherford discovered the positively charged particle at the heart of every atom, physicists are still struggling to fully understand the proton. High school physics teachers describe them as featureless balls with one unit each of positive electric charge — the perfect foils for the negatively charged electrons that buzz around them. College students learn that th
Understanding how the brain organizes and accesses spatial information — where we are, what’s around the corner, how to get there — remains an exquisite challenge. The process involves recalling an entire network of memories and stored spatial data from tens of billions of neurons, each connected to thousands of others. Neuroscientists have identified key elements such as grid cells, neurons that
In 1963, the mathematician Roy Kerr found a solution to Einstein’s equations that precisely described the space-time outside what we now call a rotating black hole. (The term wouldn’t be coined for a few more years.) In the nearly six decades since his achievement, researchers have tried to show that these so-called Kerr black holes are stable. What that means, explained Jérémie Szeftel, a mathema
June Huh often finds himself lost. Every afternoon, he takes a long walk around Princeton University, where he’s a professor in the mathematics department. On this particular day in mid-May, he’s making his way through the woods around the nearby Institute for Advanced Study — “Just so you know,” he says as he considers a fork in the path ahead, “I don’t know where we are” — pausing every so often
Leslie Lamport may not be a household name, but he’s behind a few of them for computer scientists: the typesetting program LaTeX and the work that made cloud infrastructure at Google and Amazon possible. He’s also brought more attention to a handful of problems, giving them distinctive names like the bakery algorithm and the Byzantine Generals Problem. This is no accident. The 81-year-old computer
When the mathematicians Jeff Kahn and Gil Kalai first posed their “expectation threshold” conjecture in 2006, they didn’t believe it themselves. Their claim — a broad assertion about mathematical objects called random graphs — seemed too strong, too all-encompassing, too bold to possibly be true. It felt more like wishful thinking than a reflection of mathematical truth. Even so, no one could prov
In a nearly 400-page paper posted in March, the mathematicians Mohammed Abouzaid and Andrew Blumberg of Columbia University have constructed a major extension of one of the biggest advances in geometry in recent decades. The work they built on relates to a well-known conjecture from the 1960s made by Vladimir Arnold. Arnold was studying classical mechanics and wanted to know when the orbits of pla
In a burst of work, mathematicians have moved closer to understanding just how many ways there are of contorting a simple sphere so that, in the end, it still resembles itself. Much of the progress comes from results by Tadayuki Watanabe at Kyoto University. Over the last several years, he has figured out how to adapt a powerful technique for the broader purpose of studying spheres of any number o
The infamous twin paradox sends the astronaut Alice on a blazing-fast space voyage. When she returns to reunite with her twin, Bob, she finds that he has aged much faster than she has. It’s a well-known but perplexing result: Time slows if you’re moving fast. Gravity does the same thing. Earth — or any massive body — warps space-time in a way that slows time, according to Albert Einstein’s general
In a preprint posted online Thursday night, researchers at Google in collaboration with physicists at Stanford, Princeton and other universities say that they have used Google’s quantum computer to demonstrate a genuine “time crystal.” In addition, a separate research group claimed earlier this month to have created a time crystal in a diamond. A novel phase of matter that physicists have strived
An editorially independent publication supported by the Simons Foundation.
Grade school math students are likely familiar with teachers admonishing them not to just guess the answer to a problem. But a new proof establishes that, in fact, the right kind of guessing is sometimes the best way to solve systems of linear equations, one of the bedrock calculations in math. As a result, the proof establishes the first method capable of surpassing what had previously been a har
Given that everything in the universe reduces to particles, a question presents itself: What are particles? The easy answer quickly shows itself to be unsatisfying. Namely, electrons, photons, quarks and other “fundamental” particles supposedly lack substructure or physical extent. “We basically think of a particle as a pointlike object,” said Mary Gaillard, a particle theorist at the University o
In mid-March, the mathematicians Joshua Greene and Andrew Lobb found themselves in the same situation: locked down and struggling to adjust while the COVID-19 pandemic grew outside their doors. They decided to cope by throwing themselves into their research. “I think the pandemic was really kind of galvanizing,” said Greene, a professor at Boston College. “We each decided it would be best to lean
“Physicists like to probe the extreme,” said Garrett Goon, a physicist at Carnegie Mellon University. “The fact that you can’t go further, that something is changing, something is blocking you — something interesting is happening there.” For decades, black holes have played the headlining role in the thought experiments that physicists use to probe nature’s extremes. These invisible spheres form w
In the summer of 2018, at a conference on low-dimensional topology and geometry, Lisa Piccirillo heard about a nice little math problem. It seemed like a good testing ground for some techniques she had been developing as a graduate student at the University of Texas, Austin. “I didn’t allow myself to work on it during the day,” she said, “because I didn’t consider it to be real math. I thought it
Donald Knuth is a computer scientist who came of age with his field. During the nascent years of computer programming in the middle of the last century, a candy company ran a contest that summoned his talents as a 13-year-old. The contest asked kids to determine how many words could be made from the letters of the candy’s name: Ziegler’s Giant Bar. It was a well-defined problem with distinct piece
When Andrew Wiles proved Fermat’s Last Theorem in the early 1990s, his proof was hailed as a monumental step forward not just for mathematicians but for all of humanity. The theorem is simplicity itself — it posits that xn + yn = zn has no positive whole-number solutions when n is greater than 2. Yet this simple claim tantalized legions of would-be provers for more than 350 years, ever since the F
The universe has cooked up all sorts of bizarre and beautiful forms of matter, from blazing stars to purring cats, out of just three basic ingredients. Electrons and two types of quarks, dubbed “up” and “down,” mix in various ways to produce every atom in existence. But puzzlingly, this family of matter particles — the up quark, down quark and electron — is not the only one. Physicists have discov
In 1935, Albert Einstein, working with Boris Podolsky and Nathan Rosen, grappled with a possibility revealed by the new laws of quantum physics: that two particles could be entangled, or correlated, even across vast distances. The very next year, Alan Turing formulated the first general theory of computing and proved that there exists a problem that computers will never be able to solve. These two
Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe. The post The Map of Mathematics first appeared on Quanta Magazine
Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again. The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring. “This is a really dangerous problem. People become obsessed with it and
In the lowlands of Bolivia, the most isolated of the Tsimané people live in communities without electricity; they don’t own televisions, computers or phones, and even battery-powered radios are rare. Their minimal exposure to Western culture happens mostly during occasional trips to nearby towns. To the researchers who make their way into Tsimané villages by truck and canoe each summer, that isola
The equal sign is the bedrock of mathematics. It seems to make an entirely fundamental and uncontroversial statement: These things are exactly the same. But there is a growing community of mathematicians who regard the equal sign as math’s original error. They see it as a veneer that hides important complexities in the way quantities are related — complexities that could unlock solutions to an eno
A paper posted online this month has settled a nearly 30-year-old conjecture about the structure of the fundamental building blocks of computer circuits. This “sensitivity” conjecture has stumped many of the most prominent computer scientists over the years, yet the new proof is so simple that one researcher summed it up in a single tweet. “This conjecture has stood as one of the most frustrating
Four thousand years ago, the Babylonians invented multiplication. Last month, mathematicians perfected it. On March 18, two researchers described the fastest method ever discovered for multiplying two very large numbers. The paper marks the culmination of a long-running search to find the most efficient procedure for performing one of the most basic operations in math. “Everybody thinks basically
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