The American Institute of Mathematics (AIM), one of the leading math institutes in the U.S., announced today that after four years of intensive collaboration, 18 leading mathematicians and computer scientists from the U.S. and Europe have successfully mapped E8, one of the largest and most complicated structures in mathematics. Partners on this project included MIT, Cornell University, University of Michigan, University of Utah and University of Maryland.
The E8 root system consists of 240 vectors in an eight-dimensional space. See what is E8? Those vectors are the vertices (corners) of an eight-dimensional object called the Gosset polytope 421. In the 1960s, Peter McMullen drew (by hand) a 2-dimensional representation of the Gosset polytope 421. The image shown below was computer-generated by John Stembridge, based on McMullen's drawing. Credit: AIM
The findings will be unveiled today, Monday, March 19 at 2 p.m. Eastern, at a presentation by David Vogan, Professor of Mathematics at MIT and member of the team that mapped E8. The presentation is open to the public and is taking place at MIT, Building 1, Room 190.
E8, (pronounced "E eight") is an example of a Lie (pronounced "Lee") group. Lie groups were invented by the 19th century Norwegian mathematician Sophus Lie to study symmetry. Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects. Mathematicians study symmetries in higher dimensions. In fact, E8 is the symmetries of a geometric object like a sphere, cylinder or cone, but this object is 57-dimensional. E8 is itself is 248-dimensional. For details on E8 visit http://aimath.org/E8/.
"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Jeffrey Adams, Project Leader and Mathematics Professor at the University of Maryland. "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." The mapping of E8 may well have unforeseen implications in mathematics and physics which won’t be evident for years to come.
"This is an exciting breakthrough," said Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton University and Chair of AIM's Scientific Board. "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. This project will be invaluable for future mathematicians and scientists."
Bigger than the Human Genome
The magnitude and nature of the E8 calculation invite 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. This is enough to store 45 days of continuous music in MP3-format. If written out on paper, the answer would cover an area the size of Manhattan. The computation required sophisticated new mathematical techniques and computing power not available even a few years ago. While many scientific projects involve processing large amounts of data, the E8 calculation is very different, as the size of the input is comparatively small, but the answer itself is enormous, and very dense.
"This is an impressive achievement," said 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."
According to Brian Conrey, Executive Director of the American Institute of Mathematics, "The E8 calculation is notable for both its magnitude as well as the way it was achieved. The mapping of E8 breaks the ‘mold’ of mathematicians typically known for their solitary style. People will look back on this project as a significant landmark and because of this breakthrough, mathematics will now be viewed as a team sport."
The E8 Calculation
The team that produced the E8 calculation began work four years ago. They meet together at the American Institute of Mathematics every summer, and in smaller groups throughout the year. Their work requires a mix of theoretical mathematics and intricate computer programming. According to team member David Vogan from MIT, "The literature on this subject is very dense and very difficult to understand. Even after we understood the underlying mathematics it still took more than two years to implement it on a computer." And then there came the problem of finding a computer large enough to do the calculation.
For another year, the team worked to make the calculation more efficient, so that it might fit on existing supercomputers, but it remained just beyond the capacity of the hardware available to them. The team was contemplating the prospect of waiting for a larger computer when Noam Elkies of Harvard pointed out an ingenious way to perform several small versions of the calculation, each producing an incomplete version of the answer. These incomplete answers could be assembled to give the final solution. The cost was having to run the calculation four times, plus the time to combine the answers. In the end the calculation took about 77 hours on the supercomputer Sage.
Beautiful Symmetry
At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.
Lie groups come in families. The classical groups A1, A2, A3, ... B1, B2, B3, ... C1, C2, C3, ... and D1, D2, D3, ... rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional!
To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries.
Atlas project members, Palo Alto, 2004
The Atlas of Lie Groups Project
The E8 calculation is part of an ambitious project sponsored by AIM and the National Science Foundation, 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 is the largest of the exceptional Lie groups). This is one of the most important unsolved problems of mathematics. The E8 calculation is a major step, and suggest that the Atlas team is well on the way to solving this problem. The Atlas team consists of 18 researchers from around the globe. The core group consists of Jeffrey Adams (University of Maryland), Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah) , Marc van Leeuwen (Poitiers), David Vogan (MIT), and (until his death in 2006) Fokko du Cloux (Lyon).
Written from a news release by American Institute of Mathematics.