Geometry

A new proof reveals the answer to the decades-old “moving sofa” problem. It highlights how even the simplest optimization problems can have counterintuitive answers.

A new proof marks the first progress in decades on important cases of the so-called kissing problem. Getting there meant doing away with traditional approaches.

Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.

A decade after the discovery of the “amplituhedron,” physicists have excavated more of the timeless geometry underlying the standard picture of how particles move.

A 1930s-era breakthrough is helping physicists understand how quantum threads could weave together into a holographic space-time fabric.

Mathematicians have long wondered how “shapes of constant width” behave in higher dimensions. A surprisingly simple construction has given them an answer.

Tensors are used all over math and science to reveal hidden geometric truths. What are they?

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

If you cover a surface with tiles, repetitive patterns always emerge — or do they? In this week’s episode, mathematician Natalie Priebe Frank and co-host Janna Levin discuss how recent breakthroughs in tiling can unlock structural secrets in the natural world.