The calculation is only a stepping stone, but an important one, researchers say, in a larger project to uncover subtle ways in which different equations or geometric shapes can be seen as facets of the same underlying thing—an insight that has led to some of the century's biggest discoveries in particle physics and may play a role in future theories. The result also highlights the growing trend of using computers to crack tough math problems.
The team's calculation, which took four years to prepare and three days of number-crunching on a supercomputer to finish, has produced one of the densest mathematical results in history: a table of numbers that fills 60 gigabytes of disk space. If typed out on paper, the researchers note, the results would cover the island of Manhattan.
The calculation has to do with symmetry—one of the most fundamental properties in mathematics and physics. A sphere demonstrates a simple symmetry: turn it by any amount and it appears unchanged. Underlying any symmetrical shape is a mathematical object called a Lie (pronounced "lee") group, after the 19th-century Norwegian mathematician Sophus Lie, who discovered that these groups also come into play when researchers use algebra tricks to simplify complex equations.
Lie found that the eponymous groups all belong to one of four families or one of five "exceptional" groups. The subject of the new calculation is E8, the most complex of the exceptionals, which has turned up in certain forms of string theory—the leading candidate to unify all the forces of nature into one theory.
Researchers for decades have catalogued the many "representations" of Lie groups—the ways that a group can manifest itself. The electrons in an atom, for example, exist in a cloud of probability that can be shaped like a sphere, a dumbbell or something more complicated, all of which are representations of the same Lie group. Quarks, the particles that make up protons and neutrons, obey a different Lie group.
But by the late 1980s mathematicians had started to get stuck. "People were trying to do calculations by hand and just not getting any insight," says mathematician Gregg Zuckerman of Yale University, who was not part of the new research. "I'm talking about the brightest people in the field."
So a former student of Zuckerman's, Jeffrey Adams of the University of Maryland, proposed bringing in computers. He and his colleagues started a project called the Atlas of Lie Groups and Representations. To make sure their computing tools were up to the challenge, they began tackling E8, a 248-dimensional object that describes a 57-dimensional shape in the same way that three dimensions are needed to describe a sphere, which is two-dimensional.
Team member David Vogan of the Massachusetts Institute of Technology announced there today the group's calculation, which documents the relationships between E8's many representations. "It's like the genome for E8," Adams says. "It's all of the information you need to understand E8 and its representations."
Today's announcement "will be a tremendous shot in the arm to the whole community of people" studying Lie groups, Zuckerman says. "You're really dealing with the cutting edge of pure mathematical research here."
"[I]t has been like nothing else I've ever done," Vogan wrote in an e-mail to colleagues this January after the calculation was finished.
Adams says that the challenge was finding a supercomputer with enough RAM (random-access memory) to perform the task. The group finally settled on a machine called Sage at the University of Washington, with 64 gigabytes of RAM.
The next steps, Adams says, will be to carry out equally tough calculations on each of E8's 450,000-plus representations to determine which are members of a class that are of particular interest to mathematicians.
This is the latest case in which mathematicians have relied on computers to solve thorny problems. In 2005 the Annals of Mathematics published a computer-aided proof of Kepler's conjecture (about the most efficient way to stack spheres) after reviewers spent four years checking the code fed into the computer but finally gave up without completing the task.
Experts say the Lie Atlas will be easier to confirm because mathematicians will actually use it in their work. "Here the answer itself is of great interest to experts and users, and there are many consistency checks that the answers must satisfy," says mathematician Peter Sarnak of Princeton University. "It will be interesting to see the final product, which I hope they make sure is user-friendly."