Topology

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.

Terence Tao, who has been called the “Mozart of Mathematics,” wrote an essay in 2007 about the common ingredients in “good” mathematical research. In this episode, the Fields Medalist joins Steven Strogatz to revisit the topic.

Researchers are discovering the shortest knots and fattest Möbius strips, among other “optimal shapes.”

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.