Topology

A powerful mathematical technique is used to model melting ice and other phenomena. But it has long been imperiled by certain “nightmare scenarios.” A new proof has removed that obstacle.

Is the universe flat and infinite, or something more complex? We can’t say for sure, but a new search strategy is mapping out the subtle signals that could reveal if the universe has a shape.

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.