May 21, 2013 | 13
There 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!
Now we come to Weil’s insight: given an algebraic equation, such as x2 + y2 = 1, we can look for its solutions in different domains: in the familiar numerical systems, such as real or complex numbers, or in less familiar ones, like natural numbers modulo N. For example, solutions of the above equation in real numbers form a circle, but solutions in complex numbers form a sphere. Therefore, the same equation has many avatars, just like Vishnu has 10 avatars, or incarnations, in Hinduism. The avatars of algebraic equations in complex numbers give us geometric shapes like the sphere or the surface of a donut; solutions in natural numbers modulo N give us other, more elusive, avatars. This was the main point of Weil’s Rosetta stone.
Weil used it to come up with what became known as the Weil conjectures, organizing the solutions modulo N in a way that made them look similar to geometric shapes. For instance, the surface of a donut may be covered by a mesh of circles, vertical and horizontal. Weil envisioned an analogue of this mesh for solutions modulo N. This was a stunning revelation—Weil was able to see order and harmony where others saw only chaos. The Weil conjectures represented a paradigm shift, which greatly stimulated the development of mathematics in the past 60 years. Deligne’s ingenious proof of the last and deepest of them (building on the work of his advisor Alexander Grothendieck) was one of the striking results that earned him the Abel Prize.
The Weil conjectures did for mathematics what quantum theory and Einstein’s relativity did for physics, and what the discovery of DNA did for biology. Alas, we don’t hear much about this story or about the fascinating drama of ideas unfolding in modern math. Mathematics remains, in the words of poet Hans Magnus Enzensberger, “a blind spot in our culture—alien territory, in which only the elite, the initiated few have managed to entrench themselves.” And this despite the fact that math is so deeply woven in the fabric of our lives and is becoming, more and more, the engine of our power, wealth, and technological progress.
Mathematical formulas and equations represent objective and necessary truths, which describe the world around us at the deepest level. And what’s also amazing is that we own all of them. No one can have a monopoly on mathematical knowledge; no one can claim a mathematical idea as his or her invention; no one can patent a formula. There is nothing in this world that is so deep and exquisite and yet so readily available to all. Today, our celebration of the work of a great mathematician serves as a reminder that everyone should be given equal access to this timeless and profound knowledge.
Photo credits: EEcc at Wikimedia Commons (Deligne)