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 math

By 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 timing

The 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.”

THE FREE PRESS

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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Themed popcorn buckets — more formally known as collectible concession vessels — have become big business. Special-edition snack receptacles designed for major movies sell for as much as $70.

Nicole Sperling, who covers Hollywood, reports:

All the studios and theater chains want in on them, because for somewhat inexplicable reasons, they have become astonishingly lucrative. AMC Theatres estimates that this year, buckets will generate $100 million in revenue, nearly double the level of 2023. Cinemark Theatres, the country’s third-largest chain, says bucket sales helped its merchandise revenue increase 40 percent from 2024 to 2025.

I’ll say this: They hold a lot of popcorn.

TODAY’S NUMBER

14

— That is how many Times articles the word “bollard” has appeared in this year. Can you use it in a sentence?

SPORTS

The World Cup failed to deliver the massive tourism windfall that host nations expected, though luxury travel and beer sales thrived.

Good throw: A teenage English soccer player’s long throw-in went viral, highlighting how a once-mundane part of the game is becoming a key attacking weapon. See it here.

Hari means “green” in Hindi and sees its apotheosis in hari chutney — a verdant, spicy blitz of cilantro, garlic and lemon juice. Combine that with butter and Parmesan to coat a pound of pasta, and you have hari chutney spaghetti, a candidate for your new favorite week-ending meal.

SPIRITUAL POP

Willow Smith, daughter of Will Smith and Jada Pinkett Smith, is releasing a new album this week called “The Thread.” It’s her eighth solo album in 11 years, and the second of 2026. She’s 25. “I went into this album wanting the official genre to be ‘devotional,’” she told Jon Pareles, who covers pop music. “Each and every song, I’m singing to the holy mystery and giving praise.”

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