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A Numerical Mystery From the 19th Century Finally Gets Solved

In the early 1950s, a group of researchers at the Institute for Advanced Study embarked on a high-tech project. At the behest of John von Neumann and Herman Goldstine, the physicist Hedvig Selberg programmed the IAS’s 1,700-vacuum-tube computer to calculate curious mathematical sums whose origins stretched back to the 18th century. The sums were related...

Self-Taught AI Shows Similarities to How the Brain Works

For a decade now, many of the most impressive artificial intelligence systems have been taught using a huge inventory of labeled data. An image might be labeled “tabby cat” or “tiger cat,” for example, to “train” an artificial neural network to correctly distinguish a tabby from a tiger. The strategy has been both spectacularly successful and woefully...

What Is Quantum Field Theory and Why Is It Incomplete?

Quantum field theory may be the most successful scientific theory of all time, predicting experimental results with stunning accuracy and advancing the study of higher dimensional mathematics. Yet, there’s also reason to believe that it is missing something. Steven Strogatz speaks with David Tong, a theoretical physicist at the University of Cambridge,...

Mathematicians Crack a Simple but Stubborn Class of Equations

In the third century BCE, Archimedes posed a riddle about herding cattle that, he claimed, only a truly wise person could solve. His problem ultimately boiled down to an equation that involves the difference between two squared terms, which can be written as x2 – dy2 = 1. Here, d is an integer — a positive or negative counting number — and Archimedes...

How the Physics of Nothing Underlies Everything

Millennia ago, Aristotle asserted that nature abhors a vacuum, reasoning that objects would fly through truly empty space at impossible speeds. In 1277, the French bishop Etienne Tempier shot back, declaring that God could do anything, even create a vacuum. Then a mere scientist pulled it off. Otto von Guericke invented a pump to suck the air from within...

A Biochemist’s View of Life’s Origin Reframes Cancer and Aging

All living cells power themselves by coaxing energetic electrons from one side of a membrane to the other. Membrane-based mechanisms for accomplishing this are, in a sense, as universal a feature of life as the genetic code. But unlike the genetic code, these mechanisms are not the same everywhere: The two simplest categories of cells, bacteria and...

At Long Last, Mathematical Proof That Black Holes Are Stable

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...

Geometric Analysis Reveals How Birds Mastered Flight

In a rectangular room draped in camouflage netting, four Harris’ hawks took turns flying back and forth between grass-covered perches while scientists recorded their every biomechanical flutter. The researchers were partaking of the time-honored pursuit of watching birds fly — although in this experiment, their real interest was in watching them land....

The Computer Scientist Challenging AI to Learn Better

Artificial intelligence algorithms are designed to learn in fits and starts. Instead of continuously updating their knowledge base with new information over time as humans do, algorithms can learn only during the training phase. After that, their knowledge remains frozen; they perform the task they were trained for without being able to keep learning...

Particle Physicists Puzzle Over a New Duality

Last year, the particle physicist Lance Dixon was preparing a lecture when he noticed a striking similarity between two formulas that he planned to include in his slides. The formulas, called scattering amplitudes, give the probabilities of possible outcomes of particle collisions. One of the scattering amplitudes represented the probability of two...

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