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Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

In math, sometimes the most common things are the hardest to find.

An upstart field that simplifies complex shapes is letting mathematicians understand how those shapes depend on the space in which you visualize them.

The mathematician Caucher Birkar was born on a subsistence farm and raised in the middle of the brutal war between Iran and Iraq. After fleeing to England, he has gone on to impose order on a wild landscape of mathematical equations.

Voevodsky’s friends remember him as constitutionally unable to compromise on the truth — a quality that led him to produce some of the most important mathematics of the 20th century.

June Huh thought he had no talent for math until a chance meeting with a legendary mind. A decade later, his unorthodox approach to mathematical thinking has led to major breakthroughs.

An unexpected connection has emerged between the results of physics experiments and an important, seemingly unrelated set of numbers in pure mathematics.

Three years ago, a solitary mathematician released an impenetrable proof of the famous *abc* conjecture. At a recent conference dedicated to the work, optimism mixed with bafflement.

A new breakthrough that bridges number theory and geometry is just the latest triumph for a close-knit group of mathematicians.