Proofs

Daniel Spielman solves important problems by thinking hard — about other questions.

Why verify every line of a proof, when just a few checks will do?

Leslie Lamport revolutionized how computers talk to each other. Now he’s working on how engineers talk to their machines.

Lillian Pierce wants to transform access to the world of mathematics, while making headway on problems that bridge the discrete and continuous.

A new computer program fashioned after artificial intelligence systems like AlphaGo has solved several open problems in combinatorics and graph theory.

Two recent collaborations between mathematicians and DeepMind demonstrate the potential of machine learning to help researchers generate new mathematical conjectures.

Three mathematicians show, for the first time, how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized.

Watanabe invented a new way of distinguishing shapes on his way to solving the last open case of the Smale conjecture, a central question in topology about symmetries of the sphere.

Michael Freedman’s momentous 1981 proof of the four-dimensional Poincaré conjecture was on the verge of being lost. The editors of a new book are trying to save it.