Physicists have wondered for decades whether infinitely dense points known as singularities can ever exist outside black holes, which would expose the mysteries of quantum gravity for all to see.

On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics. In mathematics, a simple problem is often not what it seems. Earlier this ...

The original version of this story appeared in Quanta Magazine. Much of mathematics is driven by intuition, by a deep-rooted sense of what should be true. But sometimes instinct can lead a ...