Geometry

The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.

A new proof extends the work of the late Maryam Mirzakhani, cementing her legacy as a pioneer of alien mathematical realms.

Physicists and marine biologists built a quantitative framework that predicts how coral polyps collectively construct a variety of coral shapes.

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.