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Proof Assistant Makes Jump to Big-League Math

Computer proof assistants have been an intriguing subplot in mathematics for years — promising to automate core aspects of the way mathematicians work, but in practice having little effect on the field. But a new result, completed in early June, has the feel of a rookie’s first hit in the big leagues: At last, a proof assistant has made a real contribution...

A Soil-Science Revolution Upends Plans to Fight Climate Change

The hope was that the soil might save us. With civilization continuing to pump ever-increasing amounts of carbon dioxide into the atmosphere, perhaps plants — nature’s carbon scrubbers — might be able to package up some of that excess carbon and bury it underground for centuries or longer. That hope has fueled increasingly ambitious climate change–mitigation...

The ‘Weirdest’ Matter, Made of Partial Particles, Defies Description

Your desk is made up of individual, distinct atoms, but from far away its surface appears smooth. This simple idea is at the core of all our models of the physical world. We can describe what’s happening overall without getting bogged down in the complicated interactions between every atom and electron. So when a new theoretical state of matter was...

How to Find Rational Points Like Your Job Depends on It

You’re sitting at the end of a long conference table, interviewing for your dream job. You’ve made it this far, but there’s just one more question you have to answer. “Is it possible for a line that passes through the origin to pass through no other rational points?” Five pairs of intense eyes watch you, waiting for your response. Do you get the job?...

Plasmid, Virus or Other? DNA ‘Borgs’ Blur Boundaries.

When the microbiologist Jillian Banfield and her colleagues started combing through samples of mud from wetland environments three years ago, they had a specific goal in mind: to recover and analyze fragments of DNA from large bacteria-killing viruses. And they did. But they also found something unexpected. Some of the DNA in those samples wasn’t immediately...

How Bell’s Theorem Proved ‘Spooky Action at a Distance’ Is Real

We take for granted that an event in one part of the world cannot instantly affect what happens far away. This principle, which physicists call locality, was long regarded as a bedrock assumption about the laws of physics. So when Albert Einstein and two colleagues showed in 1935 that quantum mechanics permits “spooky action at a distance,” as Einstein...

New Shape Opens ‘Wormhole’ Between Numbers and Geometry

The grandest project in mathematics has received a rare gift, in the form of a mammoth 350-page paper posted in February that will change the way researchers around the world investigate some of the field’s deepest questions. The work fashions a new geometric object that fulfills a bold, once fanciful dream about the relationship between geometry and...

A Video Tour of the Standard Model

Recently, Quanta has explored the collaboration between physics and mathematics on one of the most important ideas in science: quantum field theory. The basic objects of a quantum field theory are quantum fields, which spread across the universe and, through their fluctuations, give rise to the most fundamental phenomena in the physical world. We’ve...

How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said. Asperó, a mathematician at the University of East Anglia in the United Kingdom,...

The Computer Scientist Training AI to Think With Analogies

The Pulitzer Prize-winning book Gödel, Escher, Bach inspired legions of computer scientists in 1979, but few were as inspired as Melanie Mitchell. After reading the 777-page tome, Mitchell, a high school math teacher in New York, decided she “needed to be” in artificial intelligence. She soon tracked down the book’s author, AI researcher Douglas Hofstadter,...

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