Geometry

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.

Two mathematicians have proved a long-standing conjecture that is a step on the way toward finding the worst shape for packing the plane.

By studying the geometry of model space-times, researchers offer alternative views of the universe’s first moments.

Mathematicians have disproved a major conjecture about the relationship between curvature and shape.

In 1940 André Weil wrote a letter to his sister, Simone, outlining his vision for translating between three distinct areas of mathematics. Eighty years later, it still animates many of the most exciting developments in the field.

Four mathematicians broke a 75-year-old record by finding a denser way to pack high-dimensional spheres.