Combinatorics

A straightforward conjecture about runners moving around a track turns out to be equivalent to many complex mathematical questions. Three new proofs mark the first significant progress on the problem in decades.

Mathematicians are still trying to understand fundamental properties of the Fourier transform, one of their most ubiquitous and powerful tools. A new result marks an exciting advance toward that goal.

Descriptive set theorists study the niche mathematics of infinity. Now, they’ve shown that their problems can be rewritten in the concrete language of algorithms.

After more than three centuries, a geometry problem that originated with a royal bet has been solved.

The amplituhedron, a shape at the heart of particle physics, appears to be deeply connected to the mathematics of paper folding.

Fan Chung, who has an Erdős number of 1, discusses the importance of connection — both human and mathematical.

A new proof illuminates the hidden patterns that emerge when addition becomes impossible.

When pigeons outnumber pigeonholes, some birds must double up. This obvious statement — and its inverse — have deep connections to many areas of math and computer science.

By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.