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Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years. Now, Andrew Booker, a mathemat
Quantum computers are still a dream, but the era of quantum communication is here. A new experiment out of Paris has demonstrated, for the first time, that quantum communication is superior to classical ways of transmitting information. “We are the first to show a quantum advantage for transmitted information that two parties have to share to perform a useful task,” said Eleni Diamanti, an electri
When Charles Darwin articulated his theory of evolution by natural selection in On the Origin of Species in 1859, he focused on adaptations — the changes that enable organisms to survive in new or changing environments. Selection for favorable adaptations, he suggested, allowed ancient ancestral forms to gradually diversify into countless species. That concept was so powerful that we might assume
Now, after eight years of graduate school, Mahadev has succeeded. She has come up with an interactive protocol by which users with no quantum powers of their own can nevertheless employ cryptography to put a harness on a quantum computer and drive it wherever they want, with the certainty that the quantum computer is following their orders. Mahadev’s approach, Vazirani said, gives the user “levera
In a report posted online today, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a “serious, unfixable gap” within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-r
In 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours south to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin. Furey had figured out how to build on a finding of Günaydin’s from 40 years earlier — a largely forgotten result that supported a powerful suspicion about fundamental physics and
Stands for: Nondeterministic Polynomial time Short version: All problems that can be quickly verified by a classical computer once a solution is given. Precise version: A problem is in NP if, given a “yes” answer, there is a short proof that establishes the answer is correct. If the input is a string, X, and you need to decide if the answer is “yes,” then a short proof would be another string, Y,
Since then, computer scientists have hoped to contrast BQP with a class of problems known as “PH,” which encompasses all the problems workable by any possible classical computer — even unfathomably advanced ones engineered by some future civilization. Making that contrast depended on finding a problem that could be proven to be in BQP but not in PH. And now, Raz and Tal have done it. The result do
That’s a dramatic thing to say, that science has abandoned cause and effect. Isn’t that exactly what all of science is about? Of course, but you cannot see this noble aspiration in scientific equations. The language of algebra is symmetric: If X tells us about Y, then Y tells us about X. I’m talking about deterministic relationships. There’s no way to write in mathematics a simple fact — for examp
Half a century ago, the pioneers of chaos theory discovered that the “butterfly effect” makes long-term prediction impossible. Even the smallest perturbation to a complex system (like the weather, the economy or just about anything else) can touch off a concatenation of events that leads to a dramatically divergent future. Unable to pin down the state of these systems precisely enough to predict h
Paradoxically, reports of powerful quantum computations are motivating improvements to classical ones, making it harder for quantum machines to gain an advantage. “Most of the time when people talk about quantum computing, classical computing is dismissed, like something that is past its prime,” said Cristian Calude, a mathematician and computer scientist at the University of Auckland in New Zeala
After decades of heavy slog with no promise of success, quantum computing is suddenly buzzing with almost feverish excitement and activity. Nearly two years ago, IBM made a quantum computer available to the world: the 5-quantum-bit (qubit) resource they now call (a little awkwardly) the IBM Q experience. That seemed more like a toy for researchers than a way of getting any serious number crunching
In the last three decades, condensed matter physicists have discovered a wonderland of exotic new phases of matter: emergent, collective states of interacting particles that are nothing like the solids, liquids and gases of common experience. The phases, some realized in the lab and others identified as theoretical possibilities, arise when matter is chilled almost to absolute-zero temperature, hu
Mathematics is full of weird number systems that most people have never heard of and would have trouble even conceptualizing. But rational numbers are familiar. They’re the counting numbers and the fractions — all the numbers you’ve known since elementary school. But in mathematics, the simplest things are often the hardest to understand. They’re simple like a sheer wall, without crannies or ledge
Joe Nadeau, principal scientist at the Pacific Northwest Research Institute, is challenging this dogma. Random fertilization should lead to specific ratios of gene combinations in offspring, but Nadeau has found two examples just from his own lab that indicate fertilization can be far from random: Certain pairings of gamete genes are much more likely than others. After ruling out obvious alternati
In 1970, an astrophysicist named Koryo Miura conceived what would become one of the most well-known and well-studied folds in origami: the Miura-ori. The pattern of creases forms a tessellation of parallelograms, and the whole structure collapses and unfolds in a single motion — providing an elegant way to fold a map. It also proved an efficient way to pack a solar panel for a spacecraft, an idea
Even as machines known as “deep neural networks” have learned to converse, drive cars, beat video games and Go champions, dream, paint pictures and help make scientific discoveries, they have also confounded their human creators, who never expected so-called “deep-learning” algorithms to work so well. No underlying principle has guided the design of these learning systems, other than vague inspira
Scientists have been using quantum theory for almost a century now, but embarrassingly they still don’t know what it means. An informal poll taken at a 2011 conference on Quantum Physics and the Nature of Reality showed that there’s still no consensus on what quantum theory says about reality — the participants remained deeply divided about how the theory should be interpreted. Some physicists jus
In 1950, John Nash — the mathematician later featured in the book and film “A Beautiful Mind” — wrote a two-page paper that transformed the theory of economics. His crucial, yet utterly simple, idea was that any competitive game has a notion of equilibrium: a collection of strategies, one for each player, such that no player can win more by unilaterally switching to a different strategy. Nash’s eq
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As he was brushing his teeth on the morning of July 17, 2014, Thomas Royen, a little-known retired German statistician, suddenly lit upon the proof of a famous conjecture at the intersection of geometry, probability theory and statistics that had eluded top experts for decades. Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant
In the 1830s, the Irish mathematician William Rowan Hamilton reformulated Newton’s laws of motion, finding deep mathematical symmetries between an object’s position and its momentum. Then in the mid-1980s the mathematician Mikhail Gromov developed a set of techniques that transformed Hamilton’s idea into a full-blown area of mathematical research. Within a decade, mathematicians from a broad range
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In 1997, IBM’s Deep Blue system defeated the world chess champion, Garry Kasparov. At the time, the victory was widely described as a milestone in artificial intelligence. But Deep Blue’s technology turned out to be useful for chess and not much else. Computer science did not undergo a revolution. Will AlphaGo, the Go-playing system that recently defeated one of the strongest Go players in history
The occasion was a conference on the work of Shinichi Mochizuki, a brilliant mathematician at Kyoto University who in August 2012 released four papers that were both difficult to understand and impossible to ignore. He called the work “inter-universal Teichmüller theory” (IUT theory) and explained that the papers contained a proof of the abc conjecture, one of the most spectacular unsolved problem
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