P-adic numbers

The Kakeya conjecture predicts how much room you need to point a line in every direction. In one number system after another — with one important exception — mathematicians have been proving it true.

Laurent Fargues and Peter Scholze have found a new, more powerful way of connecting number theory and geometry as part of the sweeping Langlands program.

Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

The 30-year-old math sensation Peter Scholze is now one of the youngest Fields medalists for “the revolution that he launched in arithmetic geometry.”

An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.

At 28, Peter Scholze is uncovering deep connections between number theory and geometry.