An Unheralded Breakthrough: The Rosetta Stone of Mathematics

From Scientific American:

DeligneThere is no Nobel Prize in mathematics, but in 2001 the Norwegian government established a million-dollar Abel Prize, which is widely considered as an equivalent of the Nobel for mathematicians. This year’s prize was awarded to Pierre Deligne, professor emeritus at the Institute for Advanced Study in Princeton, N.J. Today, he is honored at a ceremony held in Oslo. Deligne’s most spectacular results are on the interface of two areas of mathematics: number theory and geometry. At first glance, the two subjects appear to be light-years apart. As the name suggests, number theory is the study of numbers, such as the familiar natural numbers (1, 2, 3, and so on) and fractions, or more exotic ones, such as the square root of two. Geometry, on the other hand, studies shapes, such as the sphere or the surface of a donut. But French mathematician André Weil had a penetrating insight that the two subjects are in fact closely related. In 1940, while Weil was imprisoned for refusing to serve in the army during World War II, he sent a letter to his sister Simone Weil, a noted philosopher, in which he articulated his vision of a mathematical Rosetta stone. Weil suggested that sentences written in the language of number theory could be translated into the language of geometry, and vice versa. “Nothing is more fertile than these illicit liaisons,” he wrote to his sister about the unexpected links he uncovered between the two subjects; “nothing gives more pleasure to the connoisseur.” And the key to his groundbreaking idea was something we encounter everyday when we look at the clock.

If we start working at 10:00 in the morning and work for eight hours, when do we finish? Well, 10 + 8 = 18, so a natural thing to say would be: “We finish at 18 o’clock.” This would be perfectly fine to say in France, where hours are recorded as numbers from zero to 24 (actually, not so fine, because a workday in France is usually limited to seven hours). But in the U.S. we say: “We finish at 6:00 pm.” How do we get six out of 18? We subtract 12: 18 – 12 = 6. Mathematicians call this “addition modulo 12.” Likewise, we can do addition modulo any whole number N. Just imagine a clock in which there are N hours instead of 12. For each N, we then obtain an esoteric-looking numerical system, in which we can do addition and multiplication, just like with ordinary numbers. For many years these systems looked, even to math practitioners, like something that would never have any real-world applications. In fact, English mathematician G.H. Hardy wrote, with defiance and pride, of the “uselessness” of number theory. But the joke was on him: these numerical systems are now ubiquitous in the encryption algorithms used in online banking. Every time we make a purchase online, arithmetic modulo N springs into action!

More here.