Million-Dollar Math Problem

Updated August 5, 2020 | Infoplease Staff
—By Borgna Brunner

Artist Rendering of Stardust Comet Wild-2 Encounter

After a seven-year, three-billion-mile expedition through the solar system, NASA’s Stardust spacecraft capsule landed in the Arizona desert on Jan. 15, 2006, with an impressive bounty: a canister full of tens of thousands of comet particles and a smattering of interstellar dust, the first such samples ever collected.

Stardust captured the comet particles from Comet Wild-2 (pronounced Vilt) when the spacecraft flew within 149 miles of it on Jan. 2, 2004, roughly in the vicinity of Jupiter. The spacecraft’s collector swept up particles left in the comet’s wake, preserving them in a silicon material called aerogel, which cushioned and protected the particles on their long journey to Earth.

The Stardust samples offer a time capsule to our primordial past. Comets contain some of the oldest material in the solar system, formed out of the remaining dust and gases left over after the solar system’s creation 4.6 billion years ago. Principal investigator Donald Brownlee has commented that “this has been a fantastic opportunity to collect the most primitive material in the solar system. We fully expect some of the comet particles to be older than the Sun.” Michael Zolensky, another Stardust scientist, offers a vivid sense of how intimately connected comets are to an understanding of life on Earth: “It’s like looking at our great-great grandparents. Much of Earth's water and organics—you know, the molecules in our bodies—perhaps came from comets. So these samples will tell us…basically, where our atoms and molecules came from, and how they were delivered to Earth, and in what amount.”

The samples have already defied their expectations. Some contain minerals that could only have been formed at enormously high temperatures. But comets are icy balls thought to have formed far from the Sun, in the outer, frigid regions of the solar system. Brownlee noted that “when these minerals formed they were either red-hot or white-hot grains, and yet we collected them at a comet [from] the Siberia of the solar system.” According to Zolensky, “It suggests that, if these are really from our own sun, they've been ejected out—ballistically out—all the way across the entire solar system and landed out there.…We can't give you all the answers right now. It's just great we have new mysteries to worry about now.”

About 150 scientists around the world are currently studying the comet samples, while an army of amateur scientists have turned their attention to the stardust also collected during the mission. About 65,000 volunteers from the general public will be enlisted to help find images of interstellar dust embedded in the aerogel. The volunteers of Stardust@home will search images delivered to them on the Internet, using so-called virtual microscopes. While the comet dust is visible to the naked eye, the bits of stardust are only a few microns in diameter. Just 40 to 100 grains of stardust are thought to have been collected, and searching for these particles has been described by the NASA team like “tracking down 45 ants on a football field.”

Neither Fish nor Tetrapod

The most complete fossils ever found of a transitional fish with limbs. (Credit: The Academy of Natural Sciences/Ted Daeschler)

Several times a year popular science articles breathlessly announce the discovery of a “missing link,” but in the case of the recently identified fossils of an odd creature hailing from the Canadian territory of Nunavut, this overused term is compellingly accurate. In an April 2006 issue of Nature, paleontologists revealed the discovery of a 375-million-year-old transitional species whose anatomical traits bridge the gap between fish and tetrapod (four-legged vertebrate). Nicknamed the fishapod, its formal name is Tiktaalik roseae, from the Inuit name for a large shallow-water fish.

Tiktaalik joins several other significant transitional fossils—the most famous of which is Archaeopteryx, the part-bird, part-reptile considered the “missing link” between birds and dinosaurs, which was discovered in 1860, just two years after Darwin published The Origin of Species.

The transformation of aquatic creatures into land animals took place during the Devonian period, about 410 to 356 million years ago. But before the discovery of the 375-million-year-old Tiktaalik fossils, there had been no actual fossil evidence to illustrate this crucial evolutionary moment. According to paleontologist Neil Shubin of the University of Chicago, “We are capturing a very significant transition at a key moment of time. What is significant about the animal is that it is a fossil that blurs the distinction between two forms of life—between an animal that lives in water and an animal that lives on land.”

Tiktaalik resembles a huge scaly fish with a flat, crocodile snout. What amazed scientists was its pectoral fins, which contain bones forming the beginnings of a shoulder, elbow, wrist, hand, and even nascent fingers. Shubin describes the fin as “basically a scale-covered arm,” asserting that “here’s a creature that has a fin that can do push-ups.” Tiktaalik could pull its own weight, dragging itself along in shallow water and onto dry land, much like a seal.

Tiktaalik also distinguishes itself from a fish by the existence of a primitive neck and ribs. As Harvard University paleontologist Farish A. Jenkins explains, “Out of water, these fish encountered gravitational forces very different from the relative buoyancy they enjoyed in an aquatic setting. Restructuring of the body to withstand these forces is evident in the ribs, which are plate-like and overlap like shingles, forming a rigid supporting mechanism for the trunk.” And while a fish has no need of a neck—in water, its entire torso easily falls into place behind its head when changing directions—Tiktaalik's developed neck allowed it to move its head while its body, constrained by the stronger pull of gravity on land, remained stationary. According to Edward Daeschler of the Academy of Natural Sciences, the combination of these radically new anatomical features with classic fishlike traits demonstrates that “evolution proceeds slowly…in a mosaic pattern with some elements changing while others stay the same.”

Journey to the Center of the Earth

The derrick of the JOIDES Resolution, the scientific drilling ship used to recover the ocean crust.  (Credit: IODP)

For forty years, scientists have attempted to drill deep into the ocean’s crust, an enterprise promising significant insight into the planet’s geological history. In the 1950s, the highly ambitious Mohole project sought to drill seven miles into the sea floor, all the way through the extremely dense ocean crust to the planet’s middle layer, the mantle. But the project was abandoned in 1966 as exorbitantly expensive and impractical—no drill was up to the task, and the project’s deepest hole, after nearly a decade of research, planning, and drilling, reached just 601 feet into the ocean’s crust.

But in April 2006, a team of scientists reported success on a more modest ocean drilling project, which has yielded some impressive findings. Researchers involved in the Integrated Ocean Drilling Program (IODP) managed to drill nearly a mile into the ocean crust, collecting the first intact sample ever of all the crust’s various layers, including the very deepest layer, composed of igneous rock called gabbro. Gabbro has been discovered during other ocean drilling projects—in these cases, geological disturbances had shifted the gabbro closer to the ocean’s surface. But this is the first instance in which gabbro has been found in situ.

The drilling of Hole 1256D, as the 3,796-foot-deep bore hole is unceremoniously named, took place 400 miles west of Costa Rica in the Pacific Ocean, with the 120 scientists housed aboard the research ship Joides Resolution. The location was selected because the ocean crust is thinner there than in most parts of the world. The project began in 2002 and involved three separate voyages, nearly six months of drilling at sea, and twenty-five ten-inch-wide, state-of-the-art drill bits. The Integrated Ocean Drilling Program is an international marine program spearheaded by the U.S. and Japan, and involves 18 other countries. It's considered the world's largest earth science program.

Some scientists have dubbed this intact sample the holy grail to unlocking the secrets of the ocean crust, which in turn will bring broader revelations about our planet. According to Jeff Fox, the director of IODP, “The record of the earth’s history is written in greater clarity in the sediments of rocks on the sea floor than anywhere else.”

Million-Dollar Math Problem

Jules Henri Poincare, the eminent French mathematician and natural philosopher, and one of the greatest modern astronomers

In 2000, the Clay Mathematics Institute of Cambridge, Mass., identified seven math problems it deemed the most “important classic questions that have resisted solution over the years.” Several of them had in fact resisted solution for more than a century—the Riemann Hypothesis, for example, has confounded mathematicians since its formulation in 1859. To create a bit of frisson among the public for the so-called Millennium Prize Problems, the Clay Institute announced it would offer a one-million-dollar reward apiece for solutions to the problems. While a layperson might have a tough time penetrating the quantum physics behind the “Yang-Mills existence and mass gap problem,” they have no difficulty understanding the meaning of the number 1 followed by 6 zeros and preceded by a dollar sign.

Seven years after the Clay Institute announced its challenge, the century-old Poincaré Conjecture, one of the thorniest of the millennium problems, is thought to have been solved. Ever since French mathematician Henri Poincaré posed the conjecture in 1904, at least a half-dozen eminent mathematicians—and many lesser ones—have tried and failed to crack the problem. But a series of three papers on the conjecture posted online in 2002 and 2003 by Russian Grigory Perelman have successfully withstood intense scrutiny by the mathematical community for the past four years—twice the number of years of public examination required by the Clay Institute.

Poincaré's Conjecture deals with the branch of math called topology, which is the study of shapes, spaces, and surfaces. The Clay Institute offers this deceptively friendly-sounding doughnut-and-apple explication of the bedeviling problem:

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is ‘“simply connected,”’ but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two-dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three-dimensional sphere (the set of points in four-dimensional space at unit distance from the origin).

The resolution of Poincaré's Conjecture will have enormous implications for our understanding of relativity and the shape of space.

But while mathematicians are hailing this as potentially the biggest breakthrough since Andrew Wiles solved Fermat's Last Theorem in 1994, Grigory Perelman himself has taken a decidedly standoffish attitude to his accomplishment. He has shown no interest in collecting the million-dollar prize, and instead of publishing his solution in a “refereed mathematics publication of worldwide repute,” as the Clay Institute requires, he simply published his papers online. His proof never even mentions Poincaré by name and is presented in such a sketchy and elliptical fashion that it resembles guidelines for proving the conjecture more than an actual proof. In 2006, three groups of mathematicians published papers that fill in the gaps left by Perelman's unorthodox solution. Mathematicians disagree, however, as to whether any of these papers actually add substantially to solving of the conjecture or simply explicate Perelman's work.

On Aug. 22, the International Congress of Mathematicians awarded Perelman the enormously prestigious Fields medal for his solution to the Poincaré Conjecture as well as for other significant contributions. Perelman refused to attend the conference and rejected the award. Serge Rukshin, Perelman's former teacher, described him as “a devoted scientist in the pure sense of the word. He believes that the most important thing is that the problem is solved.” Perelman hasn't entirely ruled out the Millennium Prize, however, commenting that “I’m not going to decide whether to accept the prize until it is offered.”

For commentary on Pluto's demotion from planetary status, see Pluto, Beyond Pluto, and Pluto Demoted!

Roundup of Recent Science DiscoveriesRoundup of Recent Science Discoveries, 2005
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