A computational tour de force

Last year, a mathematical calculation of the structure of E8 was unveiled by the Massachusetts Institute of Technology, amid great fanfare. E8 is a symmetrical structure discovered in 1887 by Norwegian mathematician Sophus Lie. No one thought the structure could ever be understood. E8 was mapped by an international group of 18 mathematicians, including UMass Boston Professor Alfred Noel.

Information about the accomplishment sent to journalists was embargoed until the night before the press conference. The story appeared in the usual places (Scientific American), but also in much less usual places such as the New York Times, the BBC, le Monde, and many, many others, plus generated a lively discussion in the blogsphere.

The coverage tended to focus on how the solution to a century-old mathematical problem may help solve the mysteries of the universe. Among those is the theory of everything that fully explains and links together all known physical phenomena – which physicists have sought for nearly two centuries.

“It could well be E8 that determines the deep inner structure of the universe,” said Jeffrey D. Adams, a professor of mathematics at the University of Maryland who led the project.

The publicity obscured the real breakthrough. While it was known how to do the E8 calculations in principle, they had so far been computationally intractable. The innovative large-scale computing that was key to the work likely spells the future for how longstanding math problems will be solved in the 21st century. Perhaps even more significant, the team was a rare collaboration for mathematicians who usually work alone or in small groups and rarely turn to supercomputers.

Symmetry is a fundamental principle of nature, and behind anything symmetric is a mathematical object such as a matrix. A matrix has entries of numbers in the cells formed by the rows and columns. The E8 matrix has 205 billion entries. If each entry was written in a one-inch square, the matrix would measure more than seven miles on each side.

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, "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.

A mathematician-programmer, Noel’s role within the group was to develop mathematical techniques that could be programmed on a computer. Vogan is one of Noel’s mentors.

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 a team member 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. The computation took 77 hours of computer time on the supercomputer Sage at the University of Washington.

“The E8 computation, although exceptional, is only the first step in a vast and complex program which will last for several years,” Noel said.

The effort to map E8 is part of a larger National Science Foundation sponsored project to map out all of the mathematical descriptions of symmetry for continuous objects like cones and spheres. The project is called the Atlas of Lie Groups and Representations. It has made software available both for educational and research use. The software is copyrighted by Noel and three others, and available under an open public license.

Noel's collaborator at UMass Boston, Professor Steven Jackson has become a member of the Atlas project. So far the team has developed further theories that are to be implemented in the software. Professor Jackson presented these new results at the last Atlas meeting held in March 2008 at the University of Maryland at College Park.

Before joining the Department of Mathematics at UMass in 1998, Noel was a research engineer at Peritus Software Services in Billerica and a lecturer at local colleges and universities. Noel’s research on representation theory and math education has been published in dozens of mathematics journals. Noel currently splits his time between teaching Calculus and Probability and Statistics courses at UMass and conducting research at MIT, where he is a visiting scholar.

This leaves the question of why the story of E8 took off in the press. According to Jeffrey Adams, "That is harder to understand than the polynomials for E8."

However, if this structure turns out to be fundamental to how the universe works, then it seems to indicate our universe is exceptional, and perhaps singular. But, whether this theory works perfectly or not, it is undoubtedly true that the fundamental nature of our universe can be described by mathematics.

Noel is part of UMass Boston's Computational Sciences, Analysis and Modeling research cluster. Representing an intersection of computer science, engineering, applied math, and the sciences (biology, physics, chemistry), the primary focus of the field is the construction of models and numerical analysis techniques to simulate, evaluate, and solve problems using computers.