In 1979 the Canadian-American mathematician Robert Langlands developed an ambitious and revolutionary theory that connected two branches of mathematics called number theory and group theory. In a dazzling set of conjectures and insights, the theory captured deep symmetries associated with equations that involve whole numbers, laying out what is now known as the Langlands program. Langlands knew that the task of proving the assumptions that underlie his theory would be the work of generations. But he was convinced that one stepping stone that needed confirmation dubbed the "fundamental lemma" would be reasonably straightforward. He, his collaborators and his students were able to prove special cases of this fundamental theorem. But proving the general case proved more difficult than Langlands anticipated so difficult, in fact, that it took 30 years to finally achieve. Over the past few years, Ngo Bao Chau, a Vietnamese mathematician working at Université Paris-Sud and the Institute for Advanced Study (IAS) in Princeton, formulated an ingenious proof of the fundamental lemma. When it was checked this year and confirmed to be correct, mathematicians around the globe breathed a sigh of relief. Mathematicians' work in this area in the last three decades was predicated on the principle that the fundamental lemma was indeed accurate and would one day be proved. "It's as if people were working on the far side of the river waiting for someone to throw this bridge across," says Peter Sarnak, a number theorist at IAS. "And now all of sudden everyone's work on the other side of the river has been proven."
The Top 10 Everything of 2009
- Top 10 Albums
- Top 10 Animal Stories
- Top 10 Apologies
- Top 10 Art Exhibitions
- Top 10 Awkward Moments
- Top 10 Best Business Deals
- Top 10 Breakups
- Top 10 Buzzwords
- Top 10 Children's Books
- Top 10 Crime Stories
- Top 10 Editorial Cartoons
- Top 10 Facebook Stories
- Top 10 Fashion Faux Pas
- Top 10 Fashion Moments
- Top 10 Feuds
- Top 10 Fiction Books
- Top 10 Fleeting Celebrities
- Top 10 Gadgets
- Top 10 Green Ideas
- Top 10 Heroes
- Top 10 iPhone Apps
- Top 10 Late-Night Jokes
- Top 10 Magazine Covers
- Top 10 Medical Breakthroughs
- Top 10 Movie Performances
- Top 10 Movies
- Top 10 New Species
- Top 10 News Stories
- Top 10 Nonfiction Books
- Top 10 Oddball News Stories
- Top 10 Pariahs
- Top 10 Pictures of the Year
- Top 10 Plays and Musicals
- Top 10 Political Gaffes
- Top 10 Quotes
- Top 10 Religion Stories
- Top 10 Scandals
- Top 10 Scientific Discoveries
- Top 10 Songs
- Top 10 Sports Moments
- Top 10 T-Shirt-Worthy Slogans
- Top 10 TV Ads
- Top 10 TV Episodes
- Top 10 TV Series
- Top 10 Tweets
- Top 10 Underreported Stories
- Top 10 Untruths
- Top 10 Video Games
- Top 10 Viral Videos
- Top 10 Worst Biz Deals
TIME charts the highs and lows of the past year in 50 wide-ranging lists