Mathematics

By replacing the most fundamental concept in topology, Peter Scholze and Dustin Clausen are taking the first step in a far bigger program to understand why numbers behave the way they do.

Grothendieck is revered in the world of math; outside of it, he’s known for his unusual life, if he’s known at all. But what were his actual mathematical contributions?

At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications.

A graduate student recently harnessed the complexity of mathematical proofs to create a powerful new tool in cryptography.

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it. But believing in its core principles didn’t come easily.

With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots.

AI is being used to prove new results at a rapid pace. Mathematicians think this is just the beginning.

The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean.