May 21, 2013 | 12

Edward Frenkel is Professor of Mathematics at University of California, Berkeley and author of the book

Contact Edward Frenkel via email.

Follow Edward Frenkel on Twitter as @edfrenkel. Or visit their website.

Follow Edward Frenkel on Twitter as @edfrenkel. Or visit their website.

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 *x ^{2}* +

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)

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“Deligne’s most spectacular results are on the interface of two areas of mathematics: number theory and geometry.”

I’m sure the problem is with me, but I just don’t see it. Isn’t analytic geometry (going back to the ancient greeks) an interface between numbers and geometry?

Link to thisI believe I remember reading once where one of the earliest Mathematical thinkers, I believe it was Pythagoras used purely geometrical visualizations to interpret proofs of his theories without the necessity for numerical association. It was only those less able to perform such visualizations that had to resort to numerical descriptive techniques by which they could understand and pass on his ideas.

Link to thisAnalytic geometry connects geometric shapes in a Euclidean space (such as curves on the plane) with algebraic equations. This allows one to use algebraic methods to understand various quantities related to those geometric shapes. For example, the circle of radius 1 consists of the points on the plane with coordinates x and y that satisfy the equation x squared + y squared = 1. We can use this equation to write a formula for the tangent line to the circle at any point. (This goes back to René Descartes.) This is a typical result in analytic geometry.

However, the numbers involved here (such as x and y above) are real numbers. These are the coordinates of points in the Euclidean space. In number theory, we study numbers of a different kind: natural numbers, fractions, and more general “algebraic numbers” — solutions of polynomial equations with integer coefficients such as the square root of 2 (real numbers are not considered part of number theory proper; they are part of another branch of mathematics: analysis). Nevertheless, André Weil’s showed that there are deep analogies between these “algebraic numbers” and geometric shapes such as a sphere and the surface of a donut. Deligne’s work exploited these analogies, providing insights into both number theory and geometry.

Link to thisNote: X^2 + Y^2 = 1 with X, Y complex forms a 3-sphere in R^4. Readers might be confused by the term sphere (which is usually taken to mean the 2-sphere).

Link to thisThe solutions of the equation X^2 + Y^2 = 1 with X, Y complex are in fact in one-to-one correspondence with points of the 2-sphere, minus one point. The reason the solutions form a 2-dimensional and not 3-dimensional space is that the above equation, viewed as an equation on 2 complex numbers, represents 2 equations on 4 real numbers representing the real and imaginary parts of X and Y. The first of these equations is that the real part of X^2 + Y^2 is equal to 1, and the second equation is that the imaginary part of X^2 + Y^2 is equal to 0. Because there are 2 equations, the space we get has real dimension 4-2=2. (The missing point of the sphere corresponds to the “infinite solution”, which we get in the limit when X and Y go to infinity. A more proper way to include this solution is to view this equation as an equation in the 2-dimensional complex projective space.)

Link to thisThank you! I was “seeing” the missing “modulus” signs:

|x^2| + |y^2| = 1 which, of course, weren’t there.

that was embarrassing.

Link to thisThere’s a much simpler example of the “usefulness” of modulo N arithmetic: computers operate in binary.

Programmers in the 60′s and 70′s learned how to perform mathematical operations in octal (modulo and/or hex (modulo 16) before they learned the intricacies of programming languages.

Link to thisGood point, MikeMJ! It’s funny how numbers get replaced by emoticons: the 8 in your post became a smiley face. And that’s how it should be: mathematics with a smile.

Link to thisThere’s a great video interview with Pierre Deligne over at the SImons Foundation website – http://simonsfoundation.org/science_lives_video/pierre-deligne/

Link to thisIt’s fascinating to learn of the work of a french pacifist mathematician during WWII, who turns out to have been Simone Weil’s brother. Thanks for this.

One question, though. The author of this post says that number theory and geometry seem to be light years apart. He disagrees with this proposition, but makes it sound like the realization that they are connected is a 20th century discovery. But, as some others have commented above, arguing that this ideas goes back much further, wasn’t this Descartes major insight as a mathematician back in the 1600s (wikipedia entry here http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes#Mathematical_legacy) – that one could construct what today we call a cartesian plane in order to turn equations into geometric shapes (lines, curves, circles, etc.?)

Link to this@samdiener: Thanks for your question. You are right that the links between algebra and geometry go back centuries. But, as I have already explained in a comment above (about analytic geometry and Descartes), the numbers that appear as coordinates of a Euclidean space are the so-called “real numbers”. On the other hand, number theory proper deals with “algebraic numbers” — these are the numbers that are solutions of polynomial equations whose coefficients are whole numbers. So, fractions and square root of 2 are algebraic numbers (square root of 2 solves the equation x^2=2), but pi is not (it is not a solution of any polynomial equation) – it is a real number. Of course, algebraic numbers are real numbers, but “most” real numbers are not algebraic (they are more like pi than like square root of 2).

The analogy between algebraic numbers and geometric shapes which is discussed in my article, discovered by Weil in the 1940s (though some parts of this analogy were known to Gauss and others much earlier) is much more subtle. In fact, it is not a precise statement (like the link between equations and shapes in Euclidean spaces discovered by Descartes), but only an analogy. It is achieved (according to Weil) by introducing a third realm between the two, in which we operate with whole numbers modulo N. We get then a “trilingual text”, as Weil put it in his letter to his sister (or, one might say, a kind of Rosetta stone). The three realms (or languages) are: number theory (study of algebraic numbers), “curves over finite fields” (solutions of algebraic equations in whole numbers modulo N), and geometric shapes like a sphere of the surface of a donut. Weil’s idea was that one can “translate” statements between the first to the second, and between the second to the third, even though there is no direct link between the first and the third languages. Weil’s conjectures are statements in the second language, which combined some statements that had already been known in the first and the third.

For more details, see the English translation of Weil’s letter to his sister here:

http://www.ams.org/notices/200503/fea-weil.pdf

Edward Frenkel

Link to thisI had no idea there were such huge prizes for maths.

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