Topology

For decades mathematicians have searched for a specific pair of surfaces that can’t be transformed into each other in four-dimensional space. Now they’ve found them.

Using ideas borrowed from graph theory, two mathematicians have shown that extremely complex surfaces are easy to traverse.

“Ribbon concordance” will let mathematicians compare knots by linking them across four-dimensional space.

Steven Strogatz explores the mysteries of knots with the mathematicians Colin Adams and Lisa Piccirillo.

The American mathematician invented entire new ways to understand shapes and spaces.

Mathematicians and computer scientists answered major questions in topology, set theory and even physics, even as computers continued to grow more capable.

More than 30 years ago, Andreas Floer changed geometry. Now, two mathematicians have finally figured out how to extend his revolutionary perspective.

Watanabe invented a new way of distinguishing shapes on his way to solving the last open case of the Smale conjecture, a central question in topology about symmetries of the sphere.

So-called topological quantum computing would avoid many of the problems that stand in the way of full-scale quantum computers. But high-profile missteps have led some experts to question whether the field is fooling itself.