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A simple, step-by-step breakdown of two “perfect” math proofs.

Gravitational waves have opened up new ways to test the properties of black holes — and Einstein’s theory of gravity along with them.

For decades, researchers have commonly assumed that higher oxygen levels led to the sudden diversification of animal life 540 million years ago. But one iconoclast argues the opposite: that new animal behaviors raised oxygen levels and remade the environment.

Generations of researchers have pursued his “Langlands program,” which seeks to create a grand unified theory of mathematics.

The mathematicians Günter Ziegler and Martin Aigner have spent the past 20 years collecting some of the most beautiful proofs in mathematics.

The renowned British physicist, who died at 76, left behind a riddle that could eventually lead his successors to the theory of quantum gravity.

By 1913, Albert Einstein had nearly completed general relativity. But a simple mistake set him on a tortured, two-year reconsideration of his theory. Today, mathematicians still grapple with the issues he confronted.

Researchers are building a case that long before the nervous system works, the brain sends crucial bioelectric signals to guide the growth of embryonic tissues.

The amount of energy infusing empty space seems too small to explain without a multiverse. But physicists have at least one alternative left to explore.