Topology

From brain folds to insect architecture, L. Mahadevan explains how complex biological forms and behaviors emerge through the interplay of physical forces, environment and embodiment.

Physicists recently mapped the hidden shape that underlies the quantum behaviors of a crystal, using a new method that’s expected to become ubiquitous.

A new proof represents the culmination of a 65-year-old story about anomalous shapes in special dimensions.

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.