by Amy Stewart
Those were heady times: Richard Foote, then a graduate student at the University of Cambridge and now a professor at the University of Vermont, once sat in a dank office and witnessed two famous theorists—John Thompson, now at the University of Florida, and John Conway, now at Princeton University—hashing out the details of a particularly unwieldy group. “It was amazing, like two Titans with lightning going between their brains,” Foote says. “They never seemed to be at a loss for some absolutely wonderful and totally off-the-wall techniques for doing something. It was breathtaking.”
It was during these decades that two of the proof’s biggest milestones occurred. In 1963 a theorem by mathematicians Walter Feit and John Thompson laid out a recipe for finding more simple finite groups. After that breakthrough, in 1972 Gorenstein laid out a 16-step plan for proving the Enormous Theorem—a project that would, once and for all, put all the finite simple groups in their place. It involved bringing together all the known finite simple groups, finding the missing ones, putting all the pieces into appropriate categories, and proving there could not be any others. It was big, ambitious, unruly, and, some said, implausible.
The Man with the Plan
Yet Gorenstein was a charismatic algebraist, and his vision energized a new group of mathematicians—with ambitions neither simple nor finite—who were eager to make their mark. “He was a larger-than-life personality,” says Lyons, who is at Rutgers. “He was tremendously aggressive in the way he conceived of problems and conceived of solutions. And he was very persuasive in convincing other people to help him.”
Solomon, who describes his first encounter with group theory as “love at first sight,” met Gorenstein in 1970. The National Science Foundation was hosting a summer institute on group theory at Bowdoin College, and every week mathematical celebrities were invited to the campus to give a lecture. Solomon, who was then a graduate student, remembers Gorenstein’s visit vividly. The mathematical celebrity, just arrived from his summer home on Martha’s Vineyard, was electrifying in both appearance and message.
“I’d never seen a mathematician in hot-pink pants before,” Solomon recalls.
In 1972, Solomon says, most mathematicians thought that the proof would not be done by the end of the 20th century. But within four years the end was in sight. Gorenstein largely credited the inspired methods and feverish pace of Aschbacher, who is a professor at the California Institute of Technology, for hastening the proof’s completion.
One reason the proof is so huge is that it stipulates that its list of finite simple groups is complete. That means the list includes every building block, and there are not any more. Oftentimes proving something does not exist—such as proving there cannot be any more groups—is more work than proving it does.
In 1981 Gorenstein declared the first version of the proof finished, but his celebration was premature. A problem emerged with a particularly thorny 800-page chunk, and it took some debate to resolve it successfully. Mathematicians occasionally claimed to find other flaws in the proof or to have found new groups that broke the rules. To date, those claims have failed to topple the proof, and Solomon says he is fairly confident that it will stand.
Gorenstein soon saw the theorem’s documentation for the sprawling, disorganized tangle that it had become. It was the product of a haphazard evolution. So he persuaded Lyons—and in 1982 the two of them ambushed Solomon—to help forge a revision, a more accessible and organized presentation, which would become the so-called second-generation proof. Their goals were to lay out its logic and keep future generations from having to reinvent the arguments, Lyons says. In addition, the effort would whittle the proof’s 15,000 pages down, reducing it to a mere 3,000 or 4,000.
Gorenstein envisioned a series of books that would neatly collect all the disparate pieces and streamline the logic to iron over idiosyncrasies and eliminate redundancies. In the 1980s the proof was inaccessible to all but the seasoned veterans of its forging. Mathematicians had labored on it for decades, after all, and wanted to be able to share their work with future generations. A second-generation proof would give Gorenstein a way to assuage his worries that their efforts would be lost amid heavy books in dusty libraries.
Gorenstein did not live to see the last piece put in place, much less raise a glass at the Smith and Baxter house. He died of lung cancer on Martha’s Vineyard in 1992. “He never stopped working,” Lyons recalls. “We had three conversations the day before he died, all about the proof. There were no goodbyes or anything; it was all business.”
Proving It Again
The first volume of the second-generation proof appeared in 1994. It was more expository than a standard math text and included only 2 of 30 proposed sections that could entirely span the Enormous Theorem. The second volume was published in 1996, and subsequent ones have continued to the present—the sixth appeared in 2005.
Foote says the second-generation pieces fit together better than the original chunks. “The parts that have appeared are more coherently written and much better organized,” he says. “From a historical perspective, it’s important to have the proof in one place. Otherwise, it becomes sort of folklore, in a sense. Even if you believe it’s been done, it becomes impossible to check.”
Solomon and Lyons are finishing the seventh book this summer, and a small band of mathematicians have already made inroads into the eighth and ninth. Solomon estimates that the streamlined proof will eventually take up 10 or 11 volumes, which means that just more than half of the revised proof has been published.
Solomon notes that the 10 or 11 volumes still will not entirely cover the second-generation proof. Even the new, streamlined version includes references to supplementary volumes and previous theorems, proved elsewhere. In some ways, that reach speaks to the cumulative nature of mathematics: every proof is a product not only of its time but of all the thousands of years of thought that came before.
In a 2005 article in the Notices of the American Mathematical Society, mathematician E. Brian Davies of King’s College London pointed out that the “proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual.” His article brought up the uncomfortable idea that some mathematical efforts may be too complex to be understood by mere mortals. Davies’s words drove Smith and his three coauthors to put together the comparatively concise book that was celebrated at the party in Oak Park.
The Enormous Theorem’s proof may be beyond the scope of most mathematicians—to say nothing of curious amateurs—but its organizing principle provides a valuable tool for the future. Mathematicians have a long-standing habit of proving abstract truths decades, if not centuries, before they become useful outside the field.
“One thing that makes the future exciting is that it is difficult to predict,” Solomon observes. “Geniuses come along with ideas that nobody of our generation has had. There is this temptation, this wish and dream, that there is some deeper understanding still out there.”
The Next Generation
These decades of deep thinking did not only move the proof forward; they built a community. Judith Baxter—who trained as a mathematician—says group theorists form an unusually social group. “The people in group theory are often lifelong friends,” she observes. “You see them at meetings, travel with them, go to parties with them, and it really is a wonderful community.”
Not surprisingly, these mathematicians who lived through the excitement of finishing the first iteration of the proof are eager to preserve its ideas. Accordingly, Solomon and Lyons have recruited other mathematicians to help them finish the new version and preserve it for the future. That is not easy: many younger mathematicians see the proof as something that has already been done, and they are eager for something different.
In addition, working on rewriting a proof that has already been established takes a kind of reckless enthusiasm for group theory. Solomon found a familiar devotee to the field in Capdeboscq
, one of a handful of younger mathematicians carrying the torch for the completion of the second-generation proof. She became enamored of group theory after taking a class from Solomon.
“To my surprise, I remember reading and doing the exercises and thinking that I loved it. It was beautiful,” Capdeboscq says. She got “hooked” on working on the second-generation proof after Solomon asked for her help in figuring out some of the missing pieces that would eventually become part of the sixth volume. Streamlining the proof, she says, lets mathematicians look for more straightforward approaches to difficult problems.
Capdeboscq likens the effort to refining a rough draft. Gorenstein, Lyons, and Solomon laid out the plan, but she says it is her job, and the job of a few other youngsters, to see all the pieces fall into place: “We have the road map, and if we follow it, at the end the proof should come out.”
RINKU PATEL
Bugged
FROM Popular Science
Someone once told me that a praying mantis in your home brings luck and good health. As for the one sitting on my kitchen countertop in Oakland, California, well, Jonathan Eisen certainly likes it. “That’s cool,” says the University of California at Davis microbiologist, lifting the tiny aluminum toy—with huge eyes and delicate clawlike front legs—off the cold marble. He sets it down only when something even smaller, a fruit fly, buzzes past. “Look,” he says admiringly, head cocked to my ceiling, “you have drosophila.”
Eisen is a tall guy in his 40s with a mountain-man beard, and he has shown up at my home wearing a T-shirt with sparkly pink block lettering that reads: ASK ME ABOUT FECAL TRANSPLANTS. He’s a firm believer that human health depends on bugs—not the six-legged variety, but the microbes that populate our guts and the environments in which we live, work, and play. Eisen explains that every time I open my door, a blast of air that has woven through the surrounding tree canopy carries microbes into my house—as do Amazon packages, pets, and muddy feet.
He’s musing about my oak trees when the forced-air heating clicks on. The furrows in his brow deepen. Hot, dry air shooting through a sealed house kills germs, he tells me. In fact, my whole house makes him deeply uncomfortable. It was extensively remodeled this past summer with antimicrobial fixtures, floors, and walls—now standard in many renovations. Eisen compares this practice to the overuse of antibiotics in medicine: wipe out the natural balance of good bugs, and you might not like the organisms that survive.
A mounting body of research has shown the importance of the microbes that live inside us, and scientists have been slowly cataloging species that live outside in nature. But little is known about the microbial ecosystem that surrounds us indoors, where we spend about 90 percent of our time. Recently a group of scientists, loosely connected through the Microbiology of the Built Environment Network that Eisen founded, has begun to probe it. The White House Office of Science and Technology Policy is looking into forming a national initiative to spur further research. Once we know what organisms we live with, we can begin to determine how we rely on them—and then we can tackle this question: To what extent do we need to stop protecting people from germs and instead protect germs from people?
I lead Eisen up a stairwell slathered in antimicrobial paint, and into a study with carpet treated with stain and odor guard. “You know that’s bad, right?” he asks. Then we pop into the bathroom. Eisen stares intensely at the tankless toilet. It appears to levitate off the floor like an antimicrobial spaceship. When I ask if he wants to step outside for fresh air, he looks relieved.
Charles Darwin, in On the Origin of Species, charts evolution through the Tree of Life. Its branches and roots lift some species toward fecundity while knocking others down to extinction. But Darwin’s tree didn’t include microbes, perhaps the most successful life forms of all. They make up roughly 60 percent of Earth’s biomass. There are more microbes in a teaspoon of soil than there are humans in the world.
By some measures, even we are more microbe than mammal. The trillions of microorganisms we harbor in our bodies, collectively known as our microbiome, outnumber human cells 10 to 1. Altogether, they weigh up to twice as much as the human brain, existing as a sort of sixth human superorgan whose function is linked to digesting our meals, preventing infection, and possibly even influencing our emotions and moods. Studies that describe new and essential roles for our microbiome are published almost daily. The reason for its breathtaking range is simple: our germs have evolved with us.
Microbes appear to have prospered by making themselves incredibly useful, and we’ve gladly given up space in exchange for the vitamins, digestive enzymes, and metabolites they provide. And so the discovery that the urban gut harbors up to 40 percent less microbial diversity than that of indigenous people living in a remote jungle concerns scientists. These “missing microbes,” they say, may have been decimated by several decades of industrialized foods, which limited our diets, and antibiotic use, which extended our lives at the expense of theirs.
Eisen offers another explanation for why our internal real estate might be in subprime condition: the microbiome within us depends upon the microbiome that surrounds us. “Have you seen germ-free mice?” he asks me. “They are seriously messed-up animals.” Delivered by cesarean section and raised in sterile chambers, these rodents have inflamed lungs and colons, like those seen in asthma and colitis. They’re also prone to haywire immunity and weird social tics.
Until relatively recently, sterile chambers weren’t our environments either. “We didn’t evolve in closed rooms,” says Maria Gloria Dominguez-Bello, a microbiologist at New York University who led the indigenous-microbiome study. “We evolved in nature.” Big families lived together on farms and in tenements, not exactly temples of hygiene. Livestock loped in the streets. Infectious disease rippled through cities. Roofs leaked. Sewers overflowed. Windows opened. But with modernization, we sealed ourselves away. In other words, we parted ways with the microbes that evolved with us. By redesigning our buildings, we redesigned ourselves.
Soft of heart and loud of mouth, Eisen enjoys a good jab. When I first met him at a Thai restaurant in Davis, he lifted up his shirt and stabbed himself with an insulin syringe. I flinched, but he grinned. “When I was a kid, I did this to freak people out,” he said. Now, he’s illustrating how his work in the field of microbiology is personal. Eisen has type 1 diabetes, an autoimmune disease linked to, among other things, changes in the microbiome.
To understand how seriously Eisen takes his position as the defender of microbial diversity, it’s useful to know where he got his career start: in an undergraduate internship at the D.C. Public Defender Service. It fostered a lifelong ardor for justice and an impulse to, whenever possible, stick it to the bullies. He argues that microbial communities—whether in our bodies or in buildings—function as complex ecosystems, not unlike tropical rainforests. “That doesn’t mean microbes don’t kill some people and make others sick,” he says. “But if you’re afraid of a tiger, you don’t clear-cut the rainforest. Well, you do in some cases, but that’s crazy.”
Until last year, Eisen was a member of the Forum on Microbial Threats. (He quit, saying both beneficial microbes and female scientists were underrepresented.) At the time the National Academy of Sciences first convened the forum, the prevailing narrative was that microbes were an enemy of public health and we were at war with them. The approach backfired: germs adapt to whatever drugs are thrown at them, swapping genes with neighbors to accrue antibiotic resistance. The rise of superbugs, coupled with growing awareness of the human microbiome, has led many scientists, including the forum, to rethink the merits of germ warfare.
Eisen takes a bite of stir-fry and suggests we ditch the word “pathogen” altogether. “Sometimes germs are good, sometimes they’re bad,” he says, sounding unusually Yoda-like. “Nothing is good or bad all the time.”
As someone who has spent 20 years studying microbial evolution, Eisen is in a good position to explain the paradigm shift. In 2007 he helped
launch a “genomic encyclopedia” of microbes—a splashy debut whose biggest point was all of the blank pages: we have no idea who the vast majority of our microbial neighbors are.
That hasn’t stopped us from trying to kick them out. There are now thousands of antimicrobial products on the market, which range from clothing to cutting boards. One industry report forecasts that the $1.9 billion coating market alone will more than double in 2020. Rolf Halden, an Arizona State University environmental engineer, says the marketing preys on consumers’ fears. “There’s ample evidence we use too many antimicrobials,” he says, “and without judgment.”
Halden has found that triclosan, a common antimicrobial, makes its way from products like hand soap into sewage, where it breeds antibiotic resistance. Studies have also detected high levels of triclosan in house dust. One found it counterintuitively helps staphylococcus—a common source of infection—adhere to plastic and glass surfaces. What we don’t know is how it or other antimicrobials affect the organisms that might actually help us.
This topic makes Eisen visibly agitated. He waves his fist like a trial lawyer itching to clock opposing counsel. He brings up a company hawking a new indoor sanitation technology on Twitter—a 24-hour, Purell-like system that purportedly kills everything, including Ebola. It’s an indiscrimate weapon in the old war. Struggling for composure, he says: “That doesn’t sound good.”
