Good morning. The Trump administration has imposed new tariffs on goods from more than 80 countries. The tariffs, along with the war in Iran, are threatening the U.S. economy. We have more on that below. But first: Yesterday was a big day for mathematics — particularly for four young mathematicians. I enlisted my colleague Evan Gorelick, The Morning’s best number cruncher, to explain their achievements.
Do the mathBy Evan Gorelick The Fields Medals are a quadrennial happening, like the Olympic Games and the World Cup. Each comes with 15,000 Canadian dollars (about $10,600) and a gold coin emblazoned with the bearded face of Archimedes, the Greek polymath who lived in the third century B.C. Nobody really knows what he looked like, so the artist took a swing. The medals, awarded in Philadelphia yesterday to four scholars, are sometimes called the Nobel Prizes of mathematics. But while the actual Nobels can be notoriously tardy — awarded decades after major work — the Fields are conferred at the dawn of a mathematician’s career. You have to be under 40 to be considered. That’s younger than the median age of first-time American home buyers. This year’s winners include two people born in China, one of them the third woman to receive a Fields in the 90 years they’ve existed. Another of the four medalists solved a geometry puzzle that’s a whole lot harder than it looks. The last says he’s leaving pure math and pivoting to artificial intelligence. (#awkward!) Kenneth Chang, a science reporter who has covered the Fields Medals five times since 2006, spoke to each of this year’s winners. Take a gander, and fight the urge to feel inadequate. Meet the mathletes
Yu Deng, 37, University of Chicago Maybe you remember learning about Newton’s laws of motion in school. Those laws describe the behavior of individual objects — think one billiard ball ricocheting off another. But in huge systems with billions or trillions of objects, it’s impossible to calculate the precise interactions among them. So mathematicians use statistical equations to model collective behavior. Deng managed to bridge the gap, working backward to derive one of those statistical models from Newton’s laws. It’s called the Boltzmann equation. Find out more.
John Pardon, 37, Stony Brook University Pardon works at the intersection of geometry (rigid shapes) and topology (bendy shapes). He has focused specifically on the nature of knots, which you can think of as tangled loops of rope that can only be untangled by cutting the rope. While an undergrad at Princeton, he answered a question that Mikhael Gromov, a Russian-French mathematician, had asked nearly 30 years earlier, in 1983: Can all knots, no matter how tangled, fit within a certain measure of how bendy the rope is? Pardon proved that they could not.
Jacob Tsimerman, 38, University of Toronto Tsimerman, who was born in Russia, is a number theorist. He’s best known for his work on the André-Oort conjecture, which helps describe higher-dimension objects called “Shimura varieties.” Those shapes are closely connected with “Diophantine equations,” including the one from Fermat’s last theorem: aⁿ + bⁿ = cⁿ. He admits that it’s tricky to explain what a number theorist does. But soon it won’t matter: He’s changing the focus of his research to artificial intelligence. He wants to make sure powerful A.I. models don’t run amok.
Hong Wang, 35, New York University Wang spends a lot of time thinking about long, thin tubes — not unlike uncooked spaghetti, she said. That’s because she works on a problem called the Kakeya conjecture, which ponders a set of needles pointing in all possible directions. You can’t change where needles point, but you can slide them around. Kakeya refers to Soichi Kakeya, a Japanese mathematician who started thinking about the problem in 1917. Building on his work, experts came to believe — but could never prove — that in three dimensions, the needles’ fractal dimension (a measure of pattern complexity) had to be exactly three. Wang showed last year that the conjecture is correct. Read more about her. Awkward timingThe unusual age limit on the Fields means it is both a tribute to winners’ work and a bet on their future. These are the mathematicians to watch. Right now feels like a tough time to place such a bet: A.I. is mastering the sorts of projects that math prodigies have traditionally used to launch academic careers. This past Sunday, a mathematician disproved an 87-year-old conjecture using an A.I. model from Anthropic — while simultaneously watching the World Cup final. It’s hard not to catastrophize. A group of mathematicians published a public declaration last month acknowledging the benefits of A.I. while warning that it could threaten the discipline. Tsimerman, for one, says A.I. has doubled his math productivity. “I think,” he said, “it will very shortly become more capable than mathematicians in the act of mathematics.”
Under pressure from a federal judge, the Justice Department withdrew subpoenas that had sought phone records and grand jury testimony from Times journalists who reported on security issues related to Air Force One. Judge Arun Subramanian gave the government’s lawyers a stark choice — “Either we can quash the subpoenas, or you could withdraw the subpoenas” — and then proceeded to grill them for more than an hour about their zealousness, sloppiness and the request for records from one reporter’s mother and another’s spouse. The judge’s demeanor at times resembled that of a teacher scolding his pupils. He asked few questions of The Times’s lawyers; at one point, he even apologized to one of them. “We won,” our executive editor, Joe Kahn, wrote in an email to staff last night. “We will undoubtedly face new legal challenges in this volatile era. But today is a day to celebrate the role of a free press in a robust democracy and the legal structure that protects us.”
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We’ve made these links free for you. Vows: It took him three years (and their wedding) to respond to her email. Euro summer: Visitors are changing plans because of rising temperatures. Should you skip summer in the Mediterranean? Shooting ranges: A few in Los Angeles are going upscale, with sleek interiors, plush lounges and four-figure membership fees. Some are calling them “guntry clubs.” Your pick: The most clicked story in The Morning yesterday was about killer whales making 2,000 pound fish explode.
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