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

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

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

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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

Richard Feynman looked tired when he wandered into my office. It was the end of a long, exhausting day in Santa Barbara, sometime around 1982. Events had included a seminar that was also a performance, lunchtime grilling by eager postdocs, and lively discussions with senior researchers. The life of a celebrated physicist is always intense. But our visitor still wanted to talk physics. We had a cou

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