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Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

In a major mathematical achievement, a small team of researchers has proven Zimmer’s conjecture.

New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”

A type of symmetry so unusual that it was called a “pariah” turns out to have deep connections to number theory.

To begin to understand what mathematicians and physicists see in the abstract structures of symmetries, let’s start with a familiar shape.

For centuries, mathematicians tried to solve problems by adding new values to the usual numbers. Now they’re investigating the unintended consequences of that tinkering.

The physicist-mathematician Miranda Cheng is working to harness a mysterious connection between string theory, algebra and number theory.

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