Topology

When 50 mathematicians spend a week in the woods, there’s no telling what will happen. And that’s the point.

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

Mathematicians have studied knots for centuries, but a new material is showing why some knots are better than others.

New work on the problem of “scissors congruence” explains when it’s possible to slice up one shape and reassemble it as another.

Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

A new proof shows why an uncountably infinite number of Möbius strips will never fit into a three-dimensional space.

As chemists tie the most complicated molecular knot yet, biophysicists create a “periodic table” that describes what kinds of knots are possible.

The 30-year-old math sensation Peter Scholze is now one of the youngest Fields medalists for “the revolution that he launched in arithmetic geometry.”

A complete classification could lead to a wealth of new materials and technologies. But some exotic phases continue to resist understanding.