Nature 19 March 2007; | doi:10.1038/news070319-4

Journey to the 248th dimension

Map of weird mathematical entity may point way for string theory.

John Whitfield



A map of one of the strangest and most complex entities in mathematics should be a powerful new tool for both mathematicians and physicists pursuing a unified theory of space, time and matter.

The strange 'thing' that has been mapped is a 'Lie group' called E8 — a set of maths that describes the symmetry of an (unimaginable to most) 57-dimensional object.

The creation of this map, which took 77 hours on a supercomputer, resulted in a matrix of 453,060 ? 453,060 cells, containing more than 205 billion entries — "all related in intricate and complex ways", says Jeffrey Adams, the project leader and a mathematician at the University of Maryland.

This represents 60 gigabytes of data, enough data to store 45 days of MP3 music files, or fill a piece of paper the size of Manhattan (about 60 square kilometres). The human genome takes up 1 gigabyte.

The finished product is essentially a database of information, which should come in very handy to theoretical physicists tackling grand unified theories of everything. "Now that it's done, mathematicians and physicists can use the results very easily," says Ian Stewart of the University of Warwick, UK. Adams agrees: "It's going to be a fabulous tool."

Weird exception

A Lie group is a collection of mathematical descriptors that help to illustrate the symmetry of a smooth object. The Lie group for a sphere, for example, describes all the mathematical operations that can be performed on the sphere without changing its appearance. There are an infinite number of straightforward Lie groups. But there are also five 'exceptional groups': weird one-offs of which E8, discovered in 1887, is one.

It gets stranger: E8, which represents the symmetries of a particular 57-dimensional object, has 248 dimensions itself.

"It's perhaps the most beautiful structure in all of mathematics, but it's very complex," says physicist Hermann Nicolai of the Max Planck Institute for Gravitational Physics in Potsdam, Germany.

Adams's team spent two years working out how the problem could be rendered in a form that wouldn't overwhelm the memory of a computer. The rest of the time was taken writing the code and testing the map, probing the mathematical properties of different regions to see if they provided the expected answer.

"The calculation was known to be possible in principle, but it was thought to be hopeless in practice," says Adams. "But four years ago a group of us said let's really try to do it. We're pretty sure we've got it right, but it's hard to be 100% sure."

"It's probably one of the most complicated pure mathematical calculations anyone's ever done," says Stewart. "Each entry is difficult to calculate — it's amazing they managed to do this."

Balls and string

Besides pure mathematicians, the people most familiar with E8 are physicists, and they might get the most out of the new map.

The mathematics of symmetry lies at the heart of both relativity and quantum physics. String theorists trying to unify these two areas are casting around for a type of symmetry that will let them deal with the troublesome extra dimensions thrown up by their models.

"A unified theory needs unique mathematics," says Nicolai. "What we'd like is a structure with very special properties. E8 has a flavour of this, although we don't know how the symmetry is realized in physical theory — we have to study it in more detail."

"Nobody knows what pieces of mathematics string theorists are going to need, but this will be an important piece of the toolkit," agrees Stewart. "It gives a better chance of making new and unexpected predictions."

The map will be included in the Atlas of Lie Groups and Representations, and will be available online at www.liegroups.org. The researchers also plan to publish their methodologies in a scientific journal.

New York Times, March 20, 2007

The Scientific Promise of Perfect Symmetry

By KENNETH CHANG



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 at 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,” Dr. 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,” Dr. 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 Dr. Adams’s 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,” Dr. 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,” Dr. 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,” Dr. Adams said.

有趣.没有想到E8有这么美妙复杂

我对画的那个图比较有兴趣,Fulton的表示论基本教程中 比较强调组合风格的表示论
万哲先也主持过代数学中组合方法的课题 我不知道组合结构到底有多重要的地位

组合结构在将来应该是非常重要的,组合结构运用的巧妙,往往能够带来很深刻的结果.代数几何中就有一些非常复杂的带有组合机构的定理.

E8, Platonic Solid, Octonion, Exotic Sphere