Proofs

In this special live recording from the International Congress of Mathematicians in Philadelphia, July 2026, the central question discussed was what do mathematicians really value about doing mathematics in the first place?

Mathematicians found that a broad class of networks will abruptly shift behavior past a critical point.

The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.”

AI’s greatest mathematical successes have come from answers to problems posed by a mid-20th century iconoclast. By examining what makes the Erdős problems unique, mathematicians are trying to understand how AI might change the rest of math.

When checking that solutions to certain problems are correct, it turns out, you can’t get around the inherent complexity of the quantum world.

With automated proof-checkers, a problem can be broken up into small chunks, solved bit-by-bit, then reassembled with confidence that every piece is correct. For some, this heralds a new area in mathematical research.

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.

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