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Group Effort: Mathematicians Map $$E_{8}$$

After a four-year effort, 18 mathematicians and computer scientists from the United States and Europe have mapped E8, one of the largest and most complex structures in mathematics.

David Vogan of the Massachusetts Institute of Technology announced the mathematical breakthrough on Monday, March 19, at MIT. Institutions involved in the massive computation include MIT, Cornell University, the University of Michigan, the University of Utah, and the University of Maryland.

E8 is an example of a Lie group. Nineteenth-century Norwegian mathematician Sophus Lie (1842-1899) was one of the first mathematicians to emphasize the importance of the notion of groups in geometry. He applied what are now known as Lie groups to characterize transformations and study symmetries. Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders, and cones are examples of symmetrical, three-dimensional objects. Mathematicians can study symmetries in even higher dimensions. E8 applies to a 57-dimensional mathematical structure, and its Lie algebra is 248-dimensional. A visualization of the E8 root system reveals an intricately connected pattern.

E8, which was discovered in 1887, seemed unsolvable, said Jeffrey Adams, project leader and mathematician at the University of Maryland. Hence, he noted, "this groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large scale computing to solve complicated mathematical problems."

"This is an exciting breakthrough," agreed mathematician Peter Sarnak of Princeton University. "Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science, including algebra, geometry, number theory, physics, and chemistry." At present, however, the full significance of the mapping of E8 remains unclear.

The magnitude and nature 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 E8 data, which contain all the information about E8 and its representations, are 60 times larger.

The E8 computation called for new mathematical techniques and computing power available only recently. "This is an impressive achievement," said physicist Hermann Nicolai, director of the Albert Einstein Institute in Potsdam, Germany. "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. 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."

The Atlas of Lie Groups Project

The E8 calculation is part of an ambitious project sponsored by the American Institute of Mathematics and the National Science Foundation and known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations of all the Lie groups—:E8 being the largest of the exceptional Lie groups.—H. Waldman

News Date: 
Wednesday, March 21, 2007