2002 E.W.R. Steacie Memorial Fellowship

Henri Darmon

What do theoretical mathematician Henri Darmon and online shoppers have in common?

They both appreciate the benefits of a special kind of algebraic equation called an elliptic curve. For the point-click-and-pay crowd these equations are the basis for secure online credit card transactions. For Dr. Darmon they are a portal into a realm of mathematical discovery.

The McGill University mathematician's work on elliptic curves has gained him recognition as one of the world's leading young number theorists.

It's work on mathematics' theoretical frontier for which Dr. Darmon is being awarded a 2002 Natural Sciences and Engineering Research Council (NSERC) E.W.R. Steacie Memorial Fellowship - one of Canada's premier science and engineering prizes.

Number theorists search for hidden patterns and relationships among numbers. Most basic are the whole numbers (1, 2, 3, …) that we learn as children. But number theorists also explore abstract quantities like i, the square root of -1, a number that is essential to equations that describe electricity and magnetism.

Darmon's mathematical tool of choice for finding interesting solutions to elliptic curve equations is known as complex multiplication theory.

Explored by the German mathematician Kurt Heegner in the 1950's, this tool was put on a rigorous mathematical foundation - by Princeton University mathematician Andrew Wiles - as part of his famous 1994 solution to the 350-year-old number theory riddle known as Fermat's Last Theorem.

"Elliptic curves are endowed with an extremely rich structure, which accounts for their central role in number theory," explains Dr. Darmon, who wrote one of the leading expositions of Wiles' famous proof.

What makes elliptic curves so powerful in practical and theoretical applications is what happens when you draw a line through two points on the curve, says Dr. Darmon. The line intersects the curve at a single, third point, so that new solutions to the corresponding equation can be generated from previously known ones.

The ongoing importance of elliptic curves to mathematics is highlighted by the fact that the Clay Mathematics Institute offers a million dollar prize to anyone who can prove what is known as the Birch and Swinnerton-Dyer conjecture. It posits that there should be a systematic mathematical recipe (an algorithm) for finding all the rational solutions to an elliptic curve equation.

While he's not expecting a cheque in the mail anytime soon, Dr. Darmon's research has revealed a tantalising new method for finding solutions to elliptic curve equations.

His most recent work - soon to be published in the top mathematics journal, The Annals of Mathematics - suggests that complex multiplication theory is only a part of a more general pattern. It's the first broad advance in the problem of solving elliptic curve equations since the approach of Heegner.

"My identities have been verified numerically, using a computer, in a few instances to a large degree of accuracy, so that they are true beyond a reasonable doubt, but we still seem to be very far from a proof," says Dr. Darmon. "To me this situation is profoundly exciting, because somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries."