Geometry

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.

The result could help researchers answer a larger question about flattening objects from the fourth dimension to the third dimension.

Three mathematicians show, for the first time, how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized.

The triangle angle sum theorem makes working with triangles easy. What happens when you can’t rely on it?

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.

Ana Caraiani seeks to unify mathematics through her work on the ambitious Langlands program.

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.