Mathematician helps crack conundrum

Peter Trapa has been caught up in what may be the world's biggest Lie.

Before anyone gets exercised, it should be understood that the word is pronounced Lee (after a Norwegian mathematical named Sophus Lie), that Lie groups are mathematical constructs describing a special kind of symmetry, and that the achievement by Trapa and 17 colleagues ranks as a major breakthrough in math and physics.

Using a supercomputer at the University of Washington in Seattle, they devised a mathematical description of a Lie group called E8, which Trapa said is one of the most difficult and complex of what is called the exceptional series of Lie groups.

(The formal way to print the name of this group is a capital E and a subscript 8, which doesn't reproduce in newspaper print. It is pronounced E-eight.)

To print out its solution using small type would require "a calculation the size of Manhattan," says the American Institute of Mathematics in a press release.

Trapa, associate professor of mathematics at the University of Utah, is part of a far-flung team compiling an atlas of Lie groups, which are constructs that "encode continuous symmetries."

A snowflake has sixfold symmetry because its crystal grows along hexagonal patterns. A circle has continuous symmetry. More complex figures do, too, such as a cube and certain everyday objects, as well as theoretical constructs that would exist in higher dimensions than those we experience. Lie groups are valuable to math and physics.

Members of the atlas team also worked on describing E8, which Trapa said is one of the most interesting Lie groups that humans can reasonably hope to tackle. This theoretical construct of 248 dimensions was discovered in the 19th century.

"It makes sense to take a closer look at it," Trapa said. He and his partners studied ways to make a supercomputer do the work of describing it.

The result is a matrix, a table written in square form, made of polynomials. "The size of the square table is something like 450,000 by 450,000" units, Trapa said. "So there's a lot of entries."

Commented the institute, "If each entry was written in a one inch square, then the entire matrix would measure more than 7 miles on each side."

Sage, the supercomputer that ran the calculations, produced an answer amounting to 60 gigabytes of information, the institute noted.

What good is it? The description is like a high-definition photograph of a galaxy, according to Trapa. If an astronomer wants to know more about a particular part of the galaxy, it's not hard to zoom in on that section of the photo.

With the E8 description, researchers can do the same thing, zoom in to work on new problems.