Topology

Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.

Three high schoolers and their mentor revisited a century-old theorem to prove that all knots can be found in a fractal called the Menger sponge.

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

The topologist Danny Calegari discusses the inevitability of disappointment in math, and how to learn from it.

Mathematicians have disproved a major conjecture about the relationship between curvature and shape.

Topologists prove two new results that bring some order to the confoundingly difficult study of four-dimensional shapes.

Mathematicians are using topological abstractions to find places where it’s hard to vote.

This past October, dozens of mathematicians gathered in Pasadena to create the third version of “Kirby’s list” — a compendium of the most important unsolved problems in the field.

The surprisingly subtle geometry of a familiar game shows how quickly math gets complicated.