The Numbers Behind a Fields Medalist’s Math

Evelyn Lamb:

Peter Scholze, one of the four Fields Medalists recognized at the International Congress of Mathematicians at the beginning of August, studies algebraic geometry. One of the motivating questions in that field is when there are whole number or rational number solutions to polynomial equations. For example, the equation x^2+y^2+z^2=1, which defines a sphere in three-dimensional space, has the point (3/13,4/13, 12/13) as a solution.
In his Fields Medal lecture, Scholze talked about his work in p-adic geometry. (You can read a survey article he wrote to accompany the lecture here, though frankly if you can read that, you don’t need to be reading my post about it because you’re well ahead of me in your understanding of the p-adics.) The p-adics are an alternative number system, sometimes more useful than the real numbers for tackling problems in algebraic geometry and number theory.
There are several roads that lead to the p-adics. One way to get there is to start with absolute values. We’re used to the standard absolute value, which measures a number’s distance from 0. If a number is greater than 0, its absolute value is just itself, and if a number is less than 0, its absolute value is the opposite of itself. So |1|=1 and |-1|=1.