主题：【文摘】未来25年科学界面临的125个重要课题 -- 土无尘

http://www.sciencemag.org/sciext/125th/

先转一篇不错的引言。本人理工出身，就不翻译了，免得献丑。

In Praise of Hard Questions

Tom Siegfried*

Great cases, as U.S. Supreme Court Justice Oliver Wendell Holmes suggested a century ago, may make bad law. But great questions often make very good science.

Unsolved mysteries provide science with motivation and direction. Gaps in the road to scientific knowledge are not potholes to be avoided, but opportunities to be exploited.

"Fundamental questions are guideposts; they stimulate people," says 2004 Nobel physics laureate David Gross. "One of the most creative qualities a research scientist can have is the ability to ask the right questions."

Science's greatest advances occur on the frontiers, at the interface between ignorance and knowledge, where the most profound questions are posed. There's no better way to assess the current condition of science than listing the questions that science cannot answer. "Science," Gross declares, "is shaped by ignorance."

There have been times, though, when some believed that science had paved over all the gaps, ending the age of ignorance. When Science was born, in 1880, James Clerk Maxwell had died just the year before, after successfully explaining light, electricity, magnetism, and heat. Along with gravity, which Newton had mastered 2 centuries earlier, physics was, to myopic eyes, essentially finished. Darwin, meanwhile, had established the guiding principle of biology, and Mendeleyev's periodic table--only a decade old--allowed chemistry to publish its foundations on a poster board. Maxwell himself mentioned that many physicists believed the trend in their field was merely to measure the values of physical constants "to another place of decimals."

Nevertheless, great questions raged. Savants of science debated not only the power of natural selection, but also the origin of the solar system, the age and internal structure of Earth, and the prospect of a plurality of worlds populating the cosmos.

In fact, at the time of Maxwell's death, his theory of electromagnetic fields was not yet widely accepted or even well known; experts still argued about whether electricity and magnetism propagated their effects via "action at a distance," as gravity (supposedly) did, or by Michael Faraday's "lines of force" (incorporated by Maxwell into his fields). Lurking behind that dispute was the deeper issue of whether gravity could be unified with electromagnetism (Maxwell thought not), a question that remains one of the greatest in science today, in a somewhat more complicated form.

Maxwell knew full well that his accomplishments left questions unanswered. His calculations regarding the internal motion of molecules did not agree with measurements of specific heats, for instance. "Something essential to the complete state of the physical theory of molecular encounters must have hitherto escaped us," he commented.

When Science turned 20--at the 19th century's end--Maxwell's mentor William Thomson (Lord Kelvin) articulated the two grand gaps in knowledge of the day. (He called them "clouds" hanging over physicists' heads.) One was the mystery of specific heats that Maxwell had identified; the other was the failure to detect the ether, a medium seemingly required by Maxwell's electromagnetic waves.

Filling those two gaps in knowledge required the 20th century's quantum and relativity revolutions. The ignorance enveloped in Kelvin's clouds was the impetus for science's revitalization.

Throughout the last century, pursuing answers to great questions reshaped human understanding of the physical and living world. Debates over the plurality of worlds assumed galactic proportions, specifically addressing whether Earth's home galaxy, the Milky Way, was only one of many such conglomerations of stars. That issue was soon resolved in favor of the Milky Way's nonexclusive status, in much the same manner that Earth itself had been demoted from its central role in the cosmos by Copernicus centuries before.

But the existence of galaxies outside our own posed another question, about the apparent motions of those galaxies away from one another. That issue echoed a curious report in Science's first issue about a set of stars forming a triangular pattern, with a double star at the apex and two others forming the base. Precise observations showed the stars to be moving apart, making the triangle bigger but maintaining its form.

"It seems probable that all these stars are slowly moving away from one common point, so that many years back they were all very much closer to one another," Science reported, as though the four stars had all begun their journey from the same place. Understanding such motion was a question "of the highest interest."

A half a century later, Edwin Hubble enlarged that question from one about stellar motion to the origin and history of the universe itself. He showed that galaxies also appeared to be receding from a common starting point, evidence that the universe was expanding. With Hubble's discovery, cosmology's grand questions began to morph from the philosophical to the empirical. And with the discovery of the cosmic microwave background in the 1960s, the big bang theory of the universe's birth assumed the starring role on the cosmological stage--providing cosmologists with one big answer and many new questions.

Figure 1 CREDITS: HULTON ARCHIVES/GETTY

By Science's centennial, a quarter-century ago, many gaps still remained in knowledge of the cosmos; some of them have since been filled, while others linger. At that time debate continued over the existence of planets around faraway stars, a question now settled with the discovery of dozens of planets in the solar system's galactic neighborhood. But now a bigger question looms beyond the scope of planets or even galaxies: the prospect of multiple universes, cousins to the bubble of time and space that humans occupy.

And not only may the human universe not be alone (defying the old definition of universe), humans may not be alone in their own space, either. The possible existence of life elsewhere in the cosmos remains as great a gap as any in present-day knowledge. And it is enmeshed with the equally deep mystery of life's origin on Earth.

Life, of course, inspires many deep questions, from the prospects for immortality to the prognosis for eliminating disease. Scientists continue to wonder whether they will ever be able to create new life forms from scratch, or at least simulate life's self-assembling capabilities. Biologists, physicists, mathematicians, and computer scientists have begun cooperating on a sophisticated "systems biology" aimed at understanding how the countless molecular interactions at the heart of life fit together in the workings of cells, organs, and whole animals. And if successful, the systems approach should help doctors tailor treatments to individual variations in DNA, permitting personalized medicine that deters disease without inflicting side effects. Before Science turns 150, revamped versions of modern medicine may make it possible for humans to live that long, too.

As Science and science age, knowledge and ignorance have coevolved, and the nature of the great questions sometimes changes. Old questions about the age and structure of the Earth, for instance, have given way to issues concerning the planet's capacity to support a growing and aging population.

Some great questions get bigger over time, encompassing an ever-expanding universe, or become more profound, such as the quest to understand consciousness. On the other hand, many deep questions drive science to smaller scales, more minute than the realm of atoms and molecules, or to a greater depth of detail underlying broad-brush answers to past big questions. In 1880, some scientists remained unconvinced by Maxwell's evidence for atoms. Today, the analogous debate focuses on superstrings as the ultimate bits of matter, on a scale a trillion trillion times smaller. Old arguments over evolution and natural selection have descended to debates on the dynamics of speciation, or how particular behaviors, such as altruistic cooperation, have emerged from the laws of individual competition.

Figure 2 CREDITS: ALL IMAGES HULTON ARCHIVES/GETTY EXCEPT FOR CHARLES DARWIN AND EDWIN HUBBLE, TIME-LIFE ARCHIVES/GETTY IMAGES

Great questions themselves evolve, of course, because their answers spawn new and better questions in turn. The solutions to Kelvin's clouds--relativity and quantum physics--generated many of the mysteries on today's list, from the composition of the cosmos to the prospect for quantum computers.

Ultimately, great questions like these both define the state of scientific knowledge and drive the engines of scientific discovery. Where ignorance and knowledge converge, where the known confronts the unknown, is where scientific progress is most dramatically made. "Thoroughly conscious ignorance," wrote Maxwell, "is the prelude to every real advance in science."

So when science runs out of questions, it would seem, science will come to an end. But there's no real danger of that. The highway from ignorance to knowledge runs both ways: As knowledge accumulates, diminishing the ignorance of the past, new questions arise, expanding the areas of ignorance to explore.

Maxwell knew that even an era of precision measurements is not a sign of science's end but preparation for the opening of new frontiers. In every branch of science, Maxwell declared, "the labor of careful measurement has been rewarded by the discovery of new fields of research and by the development of new scientific ideas."

If science's progress seems to slow, it's because its questions get increasingly difficult, not because there will be no new questions left to answer.

Fortunately, hard questions also can make great science, just as Justice Holmes noted that hard cases, like great cases, made bad law. Bad law resulted, he said, because emotional concerns about celebrated cases exerted pressures that distorted well-established legal principles. And that's why the situation in science is the opposite of that in law. The pressures of the great, hard questions bend and even break well-established principles, which is what makes science forever self-renewing--and which is what demolishes the nonsensical notion that science's job will ever be done.

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Is ours the only universe?

A number of quantum theorists and cosmologists are trying to figure out whether our universe is part of a bigger "multiverse." But others suspect that this hard-to-test idea may be a question for philosophers.

What drove cosmic inflation?

In the first moments after the big bang, the universe blew up at an incredible rate. But what did the blowing? Measurements of the cosmic microwave background and other astrophysical observations are narrowing the possibilities.

When and how did the first stars and galaxies form?

The broad brush strokes are visible, but the fine details aren't. Data from satellites and ground-based telescopes may soon help pinpoint, among other particulars, when the first generation of stars burned off the hydrogen "fog" that filled the universe.

Where do ultrahigh-energy cosmic rays come from?

Above a certain energy, cosmic rays don't travel very far before being destroyed. So why are cosmic-ray hunters spotting such rays with no obvious source within our galaxy?

What powers quasars?

The mightiest energy fountains in the universe probably get their power from matter plunging into whirling supermassive black holes. But the details of what drives their jets remain anybody's guess.

What is the nature of black holes?

Relativistic mass crammed into a quantum-sized object? It's a recipe for disaster--and scientists are still trying to figure out the ingredients.

Why is there more matter than antimatter?

To a particle physicist, matter and antimatter are almost the same. Some subtle difference must explain why matter is common and antimatter rare.

Does the proton decay?

In a theory of everything, quarks (which make up protons) should somehow be convertible to leptons (such as electrons)--so catching a proton decaying into something else might reveal new laws of particle physics.

What is the nature of gravity?

It clashes with quantum theory. It doesn't fit in the Standard Model. Nobody has spotted the particle that is responsible for it. Newton's apple contained a whole can of worms.

Why is time different from other dimensions?

It took millennia for scientists to realize that time is a dimension, like the three spatial dimensions, and that time and space are inextricably linked. The equations make sense, but they don't satisfy those who ask why we perceive a "now" or why time seems to flow the way it does.

Are there smaller building blocks than quarks?

Atoms were "uncuttable." Then scientists discovered protons, neutrons, and other subatomic particles--which were, in turn, shown to be made up of quarks and gluons. Is there something more fundamental still?

Are neutrinos their own antiparticles?

Nobody knows this basic fact about neutrinos, although a number of underground experiments are under way. Answering this question may be a crucial step to understanding the origin of matter in the universe.

Is there a unified theory explaining all correlated electron systems?

High-temperature superconductors and materials with giant and colossal magnetoresistance are all governed by the collective rather than individual behavior of electrons. There is currently no common framework for understanding them.

What is the most powerful laser researchers can build?

Theorists say an intense enough laser field would rip photons into electron-positron pairs, dousing the beam. But no one knows whether it's possible to reach that point.

Can researchers make a perfect optical lens?

They've done it with microwaves but never with visible light.

Is it possible to create magnetic semiconductors that work at room temperature?

Such devices have been demonstrated at low temperatures but not yet in a range warm enough for spintronics applications.

What is the pairing mechanism behind high-temperature superconductivity?

Electrons in superconductors surf together in pairs. After 2 decades of intense study, no one knows what holds them together in the complex, high-temperature materials.

Can we develop a general theory of the dynamics of turbulent flows and the motion of granular materials?

So far, such "nonequilibrium systems" defy the tool kit of statistical mechanics, and the failure leaves a gaping hole in physics.

Are there stable high-atomic-number elements?

A superheavy element with 184 neutrons and 114 protons should be relatively stable, if physicists can create it.

Is superfluidity possible in a solid? If so, how?

Despite hints in solid helium, nobody is sure whether a crystalline material can flow without resistance. If new types of experiments show that such outlandish behavior is possible, theorists would have to explain how.

What is the structure of water?

Researchers continue to tussle over how many bonds each H2O molecule makes with its nearest neighbors.

What is the nature of the glassy state?

Molecules in a glass are arranged much like those in liquids but are more tightly packed. Where and why does liquid end and glass begin?

Are there limits to rational chemical synthesis?

The larger synthetic molecules get, the harder it is to control their shapes and make enough copies of them to be useful. Chemists will need new tools to keep their creations growing.

What is the ultimate efficiency of photovoltaic cells?

Conventional solar cells top out at converting 32% of the energy in sunlight to electricity. Can researchers break through the barrier?

Will fusion always be the energy source of the future?

It's been 35 years away for about 50 years, and unless the international community gets its act together, it'll be 35 years away for many decades to come.

What drives the solar magnetic cycle?

Scientists believe differing rates of rotation from place to place on the sun underlie its 22-year sunspot cycle. They just can't make it work in their simulations. Either a detail is askew, or it's back to the drawing board.

How do planets form?

How bits of dust and ice and gobs of gas came together to form the planets without the sun devouring them all is still unclear. Planetary systems around other stars should provide clues.

What causes ice ages?

Something about the way the planet tilts, wobbles, and careens around the sun presumably brings on ice ages every 100,000 years or so, but reams of climate records haven't explained exactly how.

What causes reversals in Earth's magnetic field?

Computer models and laboratory experiments are generating new data on how Earth's magnetic poles might flip-flop. The trick will be matching simulations to enough aspects of the magnetic field beyond the inaccessible core to build a convincing case.

Are there earthquake precursors that can lead to useful predictions?

Prospects for finding signs of an imminent quake have been waning since the 1970s. Understanding faults will progress, but routine prediction would require an as-yet-unimagined breakthrough.

Is there--or was there--life elsewhere in the solar system?

The search for life--past or present--on other planetary bodies now drives NASA's planetary exploration program, which focuses on Mars, where water abounded when life might have first arisen.

What is the origin of homochirality in nature?

Most biomolecules can be synthesized in mirror-image shapes. Yet in organisms, amino acids are always left-handed, and sugars are always right-handed. The origins of this preference remain a mystery.

Can we predict how proteins will fold?

Out of a near infinitude of possible ways to fold, a protein picks one in just tens of microseconds. The same task takes 30 years of computer time.

How many proteins are there in humans?

It has been hard enough counting genes. Proteins can be spliced in different ways and decorated with numerous functional groups, all of which makes counting their numbers impossible for now.

How do proteins find their partners?

Protein-protein interactions are at the heart of life. To understand how partners come together in precise orientations in seconds, researchers need to know more about the cell's biochemistry and structural organization.

How many forms of cell death are there?

In the 1970s, apoptosis was finally recognized as distinct from necrosis. Some biologists now argue that the cell death story is even more complicated. Identifying new ways cells die could lead to better treatments for cancer and degenerative diseases.

What keeps intracellular traffic running smoothly?

Membranes inside cells transport key nutrients around, and through, various cell compartments without sticking to each other or losing their way. Insights into how membranes stay on track could help conquer diseases, such as cystic fibrosis.

What enables cellular components to copy themselves independent of DNA?

Centrosomes, which help pull apart paired chromosomes, and other organelles replicate on their own time, without DNA's guidance. This independence still defies explanation.

What roles do different forms of RNA play in genome function?

RNA is turning out to play a dizzying assortment of roles, from potentially passing genetic information to offspring to muting gene expression. Scientists are scrambling to decipher this versatile molecule.

What role do telomeres and centromeres play in genome function?

These chromosome features will remain mysteries until new technologies can sequence them.

Why are some genomes really big and others quite compact?

The puffer fish genome is 400 million bases; one lungfish's is 133 billion bases long. Repetitive and duplicated DNA don't explain why this and other size differences exist.

What is all that "junk" doing in our genomes?

DNA between genes is proving important for genome function and the evolution of new species. Comparative sequencing, microarray studies, and lab work are helping genomicists find a multitude of genetic gems amid the junk.

How much will new technologies lower the cost of sequencing?

New tools and conceptual breakthroughs are driving the cost of DNA sequencing down by orders of magnitude. The reductions are enabling research from personalized medicine to evolutionary biology to thrive.

How do organs and whole organisms know when to stop growing?

A person's right and left legs almost always end up the same length, and the hearts of mice and elephants each fit the proper rib cage. How genes set limits on cell size and number continues to mystify.

How can genome changes other than mutations be inherited?

Researchers are finding ever more examples of this process, called epigenetics, but they can't explain what causes and preserves the changes.

How is asymmetry determined in the embryo?

Whirling cilia help an embryo tell its left from its right, but scientists are still looking for the first factors that give a relatively uniform ball of cells a head, tail, front, and back.

How do limbs, fins, and faces develop and evolve?

The genes that determine the length of a nose or the breadth of a wing are subject to natural and sexual selection. Understanding how selection works could lead to new ideas about the mechanics of evolution with respect to development.

What triggers puberty?

Nutrition--including that received in utero--seems to help set this mysterious biological clock, but no one knows exactly what forces childhood to end.

Are stem cells at the heart of all cancers?

The most aggressive cancer cells look a lot like stem cells. If cancers are caused by stem cells gone awry, studies of a cell's "stemness" may lead to tools that could catch tumors sooner and destroy them more effectively.

Is cancer susceptible to immune control?

Although our immune responses can suppress tumor growth, tumor cells can combat those responses with counter-measures. This defense can stymie researchers hoping to develop immune therapies against cancer.

Can cancers be controlled rather than cured?

Drugs that cut off a tumor's fuel supplies--say, by stopping blood-vessel growth--can safely check or even reverse tumor growth. But how long the drugs remain effective is still unknown.

Is inflammation a major factor in all chronic diseases?

It's a driver of arthritis, but cancer and heart disease? More and more, the answer seems to be yes, and the question remains why and how.

How do prion diseases work?

Even if one accepts that prions are just misfolded proteins, many mysteries remain. How can they go from the gut to the brain, and how do they kill cells once there, for example.

How much do vertebrates depend on the innate immune system to fight infection?

This system predates the vertebrate adaptive immune response. Its relative importance is unclear, but immunologists are working to find out.

Does immunologic memory require chronic exposure to antigens?

Yes, say a few prominent thinkers, but experiments with mice now challenge the theory. Putting the debate to rest would require proving that something is not there, so the question likely will not go away.

Why doesn't a pregnant woman reject her fetus?

Recent evidence suggests that the mother's immune system doesn't "realize" that the fetus is foreign even though it gets half its genes from the father. Yet just as Nobelist Peter Medawar said when he first raised this question in 1952, "the verdict has yet to be returned."

What synchronizes an organism's circadian clocks?

Circadian clock genes have popped up in all types of creatures and in many parts of the body. Now the challenge is figuring out how all the gears fit together and what keeps the clocks set to the same time.

How do migrating organisms find their way?

Birds, butterflies, and whales make annual journeys of thousands of kilometers. They rely on cues such as stars and magnetic fields, but the details remain unclear.

Why do we sleep?

A sound slumber may refresh muscles and organs or keep animals safe from dangers lurking in the dark. But the real secret of sleep probably resides in the brain, which is anything but still while we're snoring away.

Why do we dream?

Freud thought dreaming provides an outlet for our unconscious desires. Now, neuroscientists suspect that brain activity during REM sleep--when dreams occur--is crucial for learning. Is the experience of dreaming just a side effect?

Why are there critical periods for language learning?

Monitoring brain activity in young children--including infants--may shed light on why children pick up languages with ease while adults often struggle to learn train station basics in a foreign tongue.

Do pheromones influence human behavior?

Many animals use airborne chemicals to communicate, particularly when mating. Controversial studies have hinted that humans too use pheromones. Identifying them will be key to assessing their sway on our social lives.

How do general anesthetics work?

Scientists are chipping away at the drugs' effects on individual neurons, but understanding how they render us unconscious will be a tougher nut to crack.

What causes schizophrenia?

Researchers are trying to track down genes involved in this disorder. Clues may also come from research on traits schizophrenics share with normal people.

What causes autism?

Many genes probably contribute to this baffling disorder, as well as unknown environmental factors. A biomarker for early diagnosis would help improve existing therapy, but a cure is a distant hope.

To what extent can we stave off Alzheimer's?

A 5- to 10-year delay in this late-onset disease would improve old age for millions. Researchers are determining whether treatments with hormones or antioxidants, or mental and physical exercise, will help.

What is the biological basis of addiction?

Addiction involves the disruption of the brain's reward circuitry. But personality traits such as impulsivity and sensation-seeking also play a part in this complex behavior.

Is morality hardwired into the brain?

That question has long puzzled philosophers; now some neuroscientists think brain imaging will reveal circuits involved in reasoning.

What are the limits of learning by machines?

Computers can already beat the world's best chess players, and they have a wealth of information on the Web to draw on. But abstract reasoning is still beyond any machine.

How much of personality is genetic?

Aspects of personality are influenced by genes; environment modifies the genetic effects. The relative contributions remain under debate.

What is the biological root of sexual orientation?

Much of the "environmental" contribution to homosexuality may occur before birth in the form of prenatal hormones, so answering this question will require more than just the hunt for "gay genes."

Will there ever be a tree of life that systematists can agree on?

Despite better morphological, molecular, and statistical methods, researchers' trees don't agree. Expect greater, but not complete, consensus.

How many species are there on Earth?

Count all the stars in the sky? Impossible. Count all the species on Earth? Ditto. But the biodiversity crisis demands that we try.

What is a species?

A "simple" concept that's been muddied by evolutionary data; a clear definition may be a long time in coming.

Why does lateral transfer occur in so many species and how?

Once considered rare, gene swapping, particularly among microbes, is proving quite common. But why and how genes are so mobile--and the effect on fitness--remains to be determined.

Who was LUCA (the last universal common ancestor)?

Ideas about the origin of the 1.5-billion-year-old "mother" of all complex organisms abound. The continued discovery of primitive microbes, along with comparative genomics, should help resolve life's deep past.

How did flowers evolve?

Darwin called this question an "abominable mystery." Flowers arose in the cycads and conifers, but the details of their evolution remain obscure.

How do plants make cell walls?

Cellulose and pectin walls surround cells, keeping water in and supporting tall trees. The biochemistry holds the secrets to turning its biomass into fuel.

How is plant growth controlled?

Redwoods grow to be hundreds of meters tall, Arctic willows barely 10 centimeters. Understanding the difference could lead to higher-yielding crops.

Why aren't all plants immune to all diseases?

Plants can mount a general immune response, but they also maintain molecular snipers that take out specific pathogens. Plant pathologists are asking why different species, even closely related ones, have different sets of defenders. The answer could result in hardier crops.

What is the basis of variation in stress tolerance in plants?

We need crops that better withstand drought, cold, and other stresses. But there are so many genes involved, in complex interactions, that no one has yet figured out which ones work how.

What caused mass extinctions?

A huge impact did in the dinosaurs, but the search for other catastrophic triggers of extinction has had no luck so far. If more subtle or stealthy culprits are to blame, they will take considerably longer to find.

Can we prevent extinction?

Finding cost-effective and politically feasible ways to save many endangered species requires creative thinking.

Why were some dinosaurs so large?

Dinosaurs reached almost unimaginable sizes, some in less than 20 years. But how did the long-necked sauropods, for instance, eat enough to pack on up to 100 tons without denuding their world?

How will ecosystems respond to global warming?

To anticipate the effects of the intensifying greenhouse, climate modelers will have to focus on regional changes and ecologists on the right combination of environmental changes.

How many kinds of humans coexisted in the recent past, and how did they relate?

The new dwarf human species fossil from Indonesia suggests that at least four kinds of humans thrived in the past 100,000 years. Better dates and additional material will help confirm or revise this picture.

What gave rise to modern human behavior?

Did Homo sapiens acquire abstract thought, language, and art gradually or in a cultural "big bang," which in Europe occurred about 40,000 years ago? Data from Africa, where our species arose, may hold the key to the answer.

What are the roots of human culture?

No animal comes close to having humans' ability to build on previous discoveries and pass the improvements on. What determines those differences could help us understand how human culture evolved.

What are the evolutionary roots of language and music?

Neuroscientists exploring how we speak and make music are just beginning to find clues as to how these prized abilities arose.

What are human races, and how did they develop?

Anthropologists have long argued that race lacks biological reality. But our genetic makeup does vary with geographic origin and as such raises political and ethical as well as scientific questions.

Why do some countries grow and others stagnate?

From Norway to Nigeria, living standards across countries vary enormously, and they're not becoming more equal.

What impact do large government deficits have on a country's interest rates and economic growth rate?

The United States could provide a test case.

Are political and economic freedom closely tied?

China may provide one answer.

Why has poverty increased and life expectancy declined in sub-Saharan Africa?

Almost all efforts to reduce poverty in sub-Saharan Africa have failed. Figuring out what will work is crucial to alleviating massive human suffering.

The following six mathematics questions are drawn from a list of seven outstanding problems selected by the Clay Mathematics Institute. (The seventh problem is discussed on p. 96.) For more details, go to www.claymath.org/millennium.

Is there a simple test for determining whether an elliptic curve has an infinite number of rational solutions?

Equations of the form y2 = x3 ax b are powerful mathematical tools. The Birch and Swinnerton-Dyer conjecture tells how to determine how many solutions they have in the realm of rational numbers--information that could solve a host of problems, if the conjecture is true.

Can a Hodge cycle be written as a sum of algebraic cycles?

Two useful mathematical structures arose independently in geometry and in abstract algebra. The Hodge conjecture posits a surprising link between them, but the bridge remains to be built.

Will mathematicians unleash the power of the Navier-Stokes equations?

First written down in the 1840s, the equations hold the keys to understanding both smooth and turbulent flow. To harness them, though, theorists must find out exactly when they work and under what conditions they break down.

Does Poincaré's test identify spheres in four-dimensional space?

You can tie a string around a doughnut, but it will slide right off a sphere. The mathematical principle behind that observation can reliably spot every spherelike object in 3D space. Henri Poincaré conjectured that it should also work in the next dimension up, but no one has proved it yet.

Do mathematically interesting zero-value solutions of the Riemann zeta function all have the form a bi?

Don't sweat the details. Since the mid-19th century, the "Riemann hypothesis" has been the monster catfish in mathematicians' pond. If true, it will give them a wealth of information about the distribution of prime numbers and other long-standing mysteries.

Does the Standard Model of particle physics rest on solid mathematical foundations?

For almost 50 years, the model has rested on "quantum Yang-Mills theory," which links the behavior of particles to structures found in geometry. The theory is breathtakingly elegant and useful--but no one has proved that it's sound.