If you thought writing calculations to describe three-dimensional objects in math class was hard, consider doing the same for one with 248 dimensions.
Mathematicians call such an object E8 (pronounced "e eight"), a symmetrical structure whose mathematical calculation has long been considered an unsolvable problem. Yet an international team of math whizzes cracked E8's symmetrical code in a large-scale computing project, which produced about 60 gigabytes of data. If they were to show their handiwork on paper, the written equation would cover an area the size of Manhattan.
David Vogan, a professor in MIT's Department of Mathematics and member of the international research team, presented the work Monday on MIT's campus. His talk was called "The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness."
Project leaders said that the work is important for several reasons. First, it brought together 18 math professors who typically work alone, in a landmark project sponsored by the National Science Foundation. Second, that large-scale computing factored heavily into solving the equation means that other difficult and long-standing math problems could be understood this way. And the work might lead to new discoveries in mathematics and physics.
"Understanding and classifying the representations of E8 …has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists," said Peter Sarnak, a professor of mathematics at Princeton University who was not involved with the work.
E8 was discovered in 1887 and it's an example of a Lie (pronounced "Lee") group. The 19th-century Norwegian mathematician Sophus Lie invented Lie groups as a way to study the symmetry of inherently symmetrical objects like the sphere. With its 248 dimensions, E8 is the largest of the higher-dimension Lie groups. Under a project called Atlas, mathematicians are trying to determine the unitary representations (or symmetries of a quantum mechanical system) of all the Lie groups.
"There are lots of ways that E8 appears in abstract mathematics, and it's going to be fun to try to find interpretations of our work in some of those appearances," said Vogan. "The uniqueness of E8 makes me hope that it should have a role to play in theoretical physics as well. So far the work in that direction is pretty speculative, but I'll stay hopeful."