Symmetry in 248 Dimensions

Institute: AIM     October 2007
A Calculation the Size of Manhattan

e8planeMathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry. The magnitude of the calculation is staggering: the answer, if written out in tiny print, would cover an area the size of Manhattan. Mathematicians are known for their solitary work style, but the assault on E8 is part of a large project bringing together 18 mathematicians from the U.S. and Europe for an intensive four-year collaboration.

“This is exciting,” said Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton University (not affiliated with the project). “Understanding and classifying the representations of Lie Groups 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 valuable for future mathematicians and scientists.”

Bigger than the Human Genome

E8 Dynkin diagramThe magnitude of the E8 calculation invites comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. That is enough space to store 45 days of continuous music in MP3 format. While many scientific projects involve processing large amounts of data, the E8 calculation is very different: the size of the input is comparatively small, but the answer itself is enormous, and very dense.

Like the Human Genome Project, these results are just the beginning. According to project leader Jeffrey Adams, “This is basic research which will have many implications, most of which we don’t understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas.” This could have unforeseen implications in mathematics and physics which do not appear for years.

According to Hermann Nicolai, a director of the Max Planck Institute in Potsdam, Germany (not affiliated with the project), “This is an impressive achievement. While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently — yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner! Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory.”

For more information visit the E8 page.