McGill Professor Wins Top Canadian Research Award

For theoretical studies on the frontier of mathematics

Ottawa, Ontario, March 5, 2002 – 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) Steacie Fellowship – one of Canada's premier science and engineering prizes.

The award is one of six announced today by Maurizio Bevilacqua, the Secretary of State for Science, Research and Development, on behalf of Allan Rock, Minister of Industry and Minister responsible for NSERC, and by Dr. Tom Brzustowski, President of NSERC.

"These winners will help Canada become one of the top five countries in the world for research and development – attracting and retaining the best and brightest minds," said Minister Rock. "This is an important part of making Canada more innovative and competitive in the global economy."

"NSERC Steacie Fellowships are awarded to Canada's most outstanding researchers," said Secretary of State Bevilacqua. "The winners continue to make a mark for themselves and for Canada on the international research scene. The NSERC Steacie will give them the opportunity and resources to develop their ideas to a new level of excellence."

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.

Dr. 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 by 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."

Contacts:

Dr. Henri Darmon, (514) 398-2263, darmon@math.mcgill.ca.

Arnet Sheppard, NSERC, (613) 995-5997, axs@nserc.ca.

The Prize

NSERC Steacie Fellowships are awarded to the most outstanding Canadian university scientists or engineers who have earned their doctorate within the last twelve years. Nominations are received by NSERC from universities across Canada and judged by a distinguished panel of independent experts. The award includes a payment to McGill University towards Dr. Darmon's salary, and increased research funding from NSERC, freeing him to pursue his research full-time. In addition, for the first time this year, the winning Steacie Fellows have been invited to compete for a special Canada Foundation for Innovation Career Award. The announcement of these awards will be made later.

The five other award winners this year are:

Canada's Innovation Strategy

On February 12, 2002, the Government of Canada launched Canada's Innovation Strategy, two papers that lay out a plan to address skills and innovation challenges for the next decade. The paper released by Minister Rock was entitled Achieving Excellence: Investing in People, Knowledge and Opportunity. It proposes goals, targets and priorities for Canada over the next decade to: create knowledge and bring ideas to market more quickly; ensure a skilled workforce in the new economy; modernize business and regulatory policies while protecting the public interest; and strengthen communities by supporting innovation at the local level. Today's announcement supports this strategy.

For more information about Canada's Innovation Strategy or to obtain a copy of either Knowledge Matters: Skills and Learning for Canadians or Achieving Excellence: Investing in People, Knowledge and Opportunity, please call 1-800-CANADA (1-800-622-6232) or visit www.innovationstrategy.gc.ca.