Topology

By replacing the most fundamental concept in topology, Peter Scholze and Dustin Clausen are taking the first step in a far bigger program to understand why numbers behave the way they do.

With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots.

The Bonnet problem asks when just a bit of information is enough to uniquely identify a whole surface.

In the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics.

Two mathematicians have proved that a straightforward question — how hard is it to untie a knot? — has a complicated answer.

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.