Geometry

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.

As topologists seek to classify shapes, the effort hinges on how to define a manifold and what it means for two of them to be equivalent.

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

Michael Freedman’s momentous 1981 proof of the four-dimensional Poincaré conjecture was on the verge of being lost. The editors of a new book are trying to save it.

How readers used their geometry skills to survive a dangerous puzzle.