*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}*y*= 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

^{2}*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)

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© 2013 ScientificAmerican.com. All rights reserved.

© 2013 ScientificAmerican.com. All rights reserved.

- Science
- Pierre Deligne
- Simone Weil