18 Mathematicians, Advanced Computers Depict the Universe Results of Computation Revealed at MIT
It is one of the most symmetrical mathematical structures in the universe.
It may underlie the Theory of Everything that physicists seek to describe the universe.
Eighteen mathematicians spent four years and 77 hours of supercomputer computation to describe this structure, with the results unveiled Monday during a talk at the Massachusetts Institute of Technology.
But it still is not easy to describe the description, at least not in words.
“It’s pretty abstract,” conceded Jeffrey D. Adams, a professor of mathematics at the University of Maryland who led the project.
For mathematicians and physicists, symmetry can provide crucial insights into a problem. A 19th-century Norwegian mathematician, Sophus Lie (rhymes with tree), wrote down what are now known as Lie groups, sets of continuous transformations — meaning the changes could be a little or a lot — that leave an object unchanged in appearance.
For example, rotate a sphere any distance around any axis, and the sphere looks exactly the same.
Later mathematicians found five exceptions to the four classes of Lie groups that Lie knew about. The most complicated of the “exceptional simple Lie groups” is E8. It describes the symmetries of a 57-dimensional object that can in essence be rotated in 248 ways without changing its appearance.
Why are there five exceptional Lie groups? “It’s just one of the beautiful magical things that happen in mathematics,” Adams said.
“You can’t really picture it,” Brian Conrey, executive director of the American Institute of Mathematics, said of E8. The institute sponsored the project with financing from the National Science Foundation.
“It’s some sort of curvy, torus type of thing,” Conrey said. “Now you start to move it around in different ways. It’s an amazingly symmetric group.”
To understand using E8 in all its possibilities requires calculation of 200 billion numbers. That is what Adams’ team did, a rare collaboration for mathematicians who usually work alone or in small groups and rarely turn to supercomputers.
Robert L. Bryant, a mathematician at Duke who was not involved in the project, gave a biological analogy. Scientists can learn a lot about an animal from its DNA, but to understand it fully “you have to grow the organism and then study it,” Bryant said. “In a certain sense, that is what the E8 team did. They used massive computation to fully develop the group E8 and its representations so that they could list its important features.”
One eventual use could be understanding the universe, another example of physics taking advantage of abstract math. Isaac Newton invented calculus to study the motion of objects. Fourier analysis, the mathematics of periodic patterns, proved essential in studying phenomena like light waves, and physicists have employed Lie groups in quantum mechanics and relativity.
“All of the physics of the 20th century is tied up with this language,” Conrey said.
E8 is the Lie group underlying some superstring theories that physicists are pursuing in an effort to tie gravity and the other fundamental forces of the universe into one theory.
“It could well be E8 that determines the deep inner structure of the universe,” Adams said.