Kevin Hartnett

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.

What’s easy for a computer to do, and what’s almost impossible? Those questions form the core of computational complexity. We present a map of the landscape.

Zeta values seem to connect distant geometric worlds. In a new proof, mathematicians finally explain why.

Generations of researchers have pursued his “Langlands program,” which seeks to create a grand unified theory of mathematics.

By 1913, Albert Einstein had nearly completed general relativity. But a simple mistake set him on a tortured, two-year reconsideration of his theory. Today, mathematicians still grapple with the issues he confronted.

Even with no one in charge, army ants work collectively to build bridges out of their bodies. New research reveals the simple rules that lead to such complex group behavior.

A famously difficult mathematical problem resisted solution for over 40 years. Mathematicians have finally resolved it by following an intuition that links number theory to physics.

A new version of AlphaGo needed no human instruction to figure out how to clobber the best Go player in the world—itself.

Voevodsky’s friends remember him as constitutionally unable to compromise on the truth—a quality that led him to produce some of the most important mathematics of the 20th century.

The color of LED lights is controlled by a clumsy process. A new mathematical discovery may make it easier for us to get the hues we want.

It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.