Mathematics

How hard is it to prove that problems are hard to solve? Meta-complexity theorists have been asking questions like this for decades. A string of recent results has started to deliver answers.

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.

A new algorithm brings together the advantages of randomness and deterministic processes to reliably construct large prime numbers.

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.