Mathematics

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.

Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.

For a half century, mathematicians have tried to define the exact circumstances under which a black hole is destined to exist. A new proof shows how a cube can help answer the question.

Mathematicians thought they were on the cusp of proving a conjecture about the ancient structures known as Apollonian circles. But a summer project would lead to its downfall.

The numbers count a variety of seemingly unrelated mathematical structures.

The proof establishes new conditions that cause connected oscillators to sway in sync.

Knowing a little about the local connections on flight maps and other networks can reveal a lot about a system’s global structure.

A new result shows how much of the plane can be colored by points that are never exactly one unit apart.