Image: Science cover
Cameron Slayden  /  Science
To prove the Poincare Conjecture, Russian mathematician Grigori Perelman used the equations for Ricci flow — a procedure for transforming irregular spaces into uniform ones. In the example shown on Science's cover, the equations say negatively curved regions (blue) must expand while positively curved regions (red) contract. Over time, the original dumbbell-shaped surface evolves into a sphere.
By Alan Boyle Science editor
updated 12/21/2006 9:28:08 PM ET 2006-12-22T02:28:08

A controversial proof of a 102-year-old mathematical puzzler has taken the top spot on the journal Science's annual list of scientific breakthroughs.

Other entries on the top 10 list ranged from the deciphering of Neanderthal DNA to studies of the world's shrinking ice sheets. But it was Russian mathematician Grigory Perelman's proof of the Poincare Conjecture , literally one of the field's million-dollar challenges, that most impressed America's premier peer-reviewed science publication.

For mathematicians, Perelman's achievement would qualify "at least as the Breakthrough of the Decade," Science's editors said.

Science's editor-in-chief, Donald Kennedy, admitted that mathematical papers don't often get the star treatment, "partly because higher mathematics is a subject that’s technically difficult to explain."

"It's difficult even for this poor biologist to understand fully," Kennedy told

However, the Russian mathematician's work on the Poincare Conjecture stands out for several reasons. First of all, Kennedy noted that the conjecture has "defeated many brilliant minds" since it was first proposed in 1904 by Henri Poincare, who is generally regarded as the founder of topology.

Rubber-sheet geometry
Topology, also known as "rubber-sheet geometry," focuses on the properties of surfaces that are preserved despite any amount of stretching or poking. To a topologist, there's no difference between a doughnut and a coffee cup, because they both have exactly one hole poking through their surfaces. But there's a big difference between a beach ball (no holes) and an inner tube (one hole).

In simple terms, Poincare proposed that there was no way to transform a no-hole surface into a one-hole surface without somehow tearing it, and that any no-hole surface can be stretched into a sphere. By the early 1980s, mathematicians were able to prove that the conjecture held true for any dimensional space — except for three-dimensional space.

To get to that last step, Perelman created a fresh mathematical vocabulary to describe the problem, then broke it down into steps that could be addressed one by one.

Science's editors said Perelman's hundreds of pages of analysis addressed the "indigestible seed at the core of topology," and could open the way to much broader breakthroughs, such as a "periodic table" of three-dimensional spaces that could do for mathematicians what the periodic table of the elements has done for chemists.

Adding to the buzz
The math alone made Perelman's achievement notable, and the circumstances surrounding his proof only added to the buzz: Starting in 2002, Perelman released the proof as a series of three papers, posted freely on the Internet for review. Other mathematicians helped fill in gaps in the step-by-step argument — and this year, the International Mathematical Union finally decided to give Perelman its highest honor, the Fields Medal.

Perelman created a stir when he refused the medal , reportedly saying that he felt isolated from the rest of the mathematical community and did not want to be seen as its "figurehead." An article about the controversy in The New Yorker added to the mystery — with Perelman complaining about lapses in the "ethical standards" of his colleagues, and other mathematicians squabbling over credit for the proof.

If the proof withstands scrutiny for another couple of years, Perelman could become eligible for a $1 million prize from the Clay Mathematics Institute — though it's not clear whether he would accept that prize, either. Meanwhile, the controversy continues to simmer. "It's clear that mathematicians regard claims as something to attack," Science's Kennedy said.

As for the practical applications of the Poincare proof, Kennedy acknowledged that there weren't any in sight. "I'm tempted to say the same thing about a lot of the exciting cosmology that has been on the menu on the last five years," he said. "We don’t have a practical application for dark energy either. But it's important for our constant probing of our understanding of the world."

Science's editors selected nine runners-up for "Breakthrough of the Year" — scientific advances that were likely to change the social and technological landscape in the years ahead.

Although the journal did not list the nine also-rans in order of importance, Kennedy gave an extra nod to findings about shrinking ice sheets on Greenland and Antarctica, in the context of the larger political debate over the effects of global climate change.

"One of the things that people are finally beginning to pay attention to in the United States is global warming," Kennedy said. Former Vice President Al Gore's documentary on the subject, "An Inconvenient Truth," has likely contributed to that extra public attention, he said.

These nine achievements filled out Science's Top 10 list of achievements for 2006:

  • Fossil DNA: Using new techniques for decoding and analyzing DNA, researchers captured genetic information from Neanderthal and mammoth fossils.
  • Shrinking ice: Researchers documented a disturbing trend this year. The Antarctic and Greenland ice sheets are losing ice, at an ever faster rate, to the oceans.
  • Transitional fish: The discovery of a fossil fish with sturdy, jointed fins made a big splash in 2006. The fish is the closest known kin to limbed vertebrates and provides a window into how life left the oceans and ventured onto terra firma.
  • Invisibility cloaks: Though it looks nothing like Harry Potter’s magical cape, the invisibility "cloak" that scientists developed this year is the first rudimentary device for shielding objects from view. The device guides incoming microwaves in such a way that they produce neither a reflection nor a shadow.
  • Hope for aging eyes: Researchers who study the form of vision loss known as age-related macular degeneration showed that the drug ranibimuzab improves vision in some patients and identifying several genes that influence a person’s susceptibility to the disease.
  • Split happens: From beach mice, to fruit flies, to butterflies , a variety of animals helped scientists uncover genetic changes that lead to the evolution of a new species.
  • Microscopic frontier: This year, biologists used new microscopy techniques that enabled them to see details smaller than about 200 nanometers, giving them a clearer view of the fine structure of cells and proteins.
  • How memories are made: Several discoveries in 2006 brought neuroscientists closer to understanding how the brain records new memories. The so-called "long-term potentiation" process that strengthens connections between neurons now seems even more likely to be the basis for remembering.
  • New breed of RNA: Scientists discovered a new class of small RNA molecules that shut down gene expression, called "Piwi-interacting RNAs."

Science took an introspective turn in selecting the stem-cell fraud perpetrated by South Korean researcher Woo-Suk Hwang and his colleagues as its "Breakdown of the Year." The false claims were contained in two papers published by Science, which had to be retracted after an investigation. The scandal led to a criminal investigation of Hwang and changes in the way Science handles papers submitted for publication.

Kennedy admitted that the case left him and his fellow editors with "just a little bit of egg on our face," but voiced hope that research into human embryonic stem cells would continue to make progress toward true breakthroughs.

"The good news is that the field is very healthy and extremely promising, and this hasn’t discouraged anybody," he said.

Finally, Science's editors listed six research areas to watch in 2007:

  • Probes for other planets should make news. The spacecraft range from the Mars Reconnaissance Orbiter and Venus Express to Europe's COROT satellite, which will look for planets around other stars. NASA's New Horizons, a probe bound for Pluto, is due to fly by Jupiter in February.
  • Hominid skulls and bones recently unearthed in the Georgian Republic, Kenya and China could provide new information about our early ancestors.
  • Primate genomes could show how evolutionary paths diverged millions of years ago. The first drafts of the galago, tree shrew and mouse lemur genomes are expected in 2007 — along with higher-resolution analyses of gorilla and orangutan genomes. "If things go as planned, a comparative analysis of all these genomes might finally begin to explain what sets humans apart," the editors write.
  • Climate change is likely to take the spotlight again in February, when the Intergovernmental Panel on Climate Change is due to issue its latest report. With control of Congress switching from the GOP to the Democrats, the global warming issue may be seen in a new light.
  • Whole-genome association studies compare the genomes of healthy people to those of the sick, potentially resulting in new genetic-based strategies for fighting disease.
  • Optical lattices, created from ultracold atoms and laser light, could soon help scientists crack problems such as high-temperature superconductivity. "Look for rapid progress in this burgeoning effort," the editors say.

For more on Science's breakthroughs of the year, check the journal's special online collection. Free registration is required. Science is published by the American Association for the Advancement of Science, which has a content partnership with

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