Topology

High-dimensional spheres can have a much wider variety of structures than mathematicians thought possible.

Eugenio Calabi, who died on September 25, conceived of novel geometric objects that later became fundamental to string theory.

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.

By treating Earth as a topological insulator — a state of quantum matter — physicists found a powerful explanation for the movements of the planet’s air and seas.

A series of new papers describes how to fully characterize key dynamical systems with relatively little data.

In all bilaterally symmetrical animals, from humans down to simple worms, nerves cross from one side of the body to the opposite side of the brain. Geometry may explain why.

So-called “higher symmetries” are illuminating everything from particle decays to the behavior of complex quantum systems.

The prize-winning geometer feels most fulfilled when exploring the fertile ground where constraint meets creation.

Can mathematics handle things that are essentially the same without being exactly equal? Category theorist Eugenia Cheng and host Steven Strogatz discuss the power and pleasures of abstraction.