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Maths boffins solve 248 dimensional problem

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This is a picture of of the 248-dimensional Lie algebra of E8, credit AIM

Physicists working to unify the mutually contradictory theories of quantum mechanics and gravity could have a new tool at their disposal, thanks to a group of mathematicians who met at the American Institute of Mathematics.

The researchers have mapped a thing called the E8 Lie Group (pronounced Lee, after mathematician Sophus Marius Lie).

The researchers behind the work explain in a press announcement:

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.  

At its most basic, a Lie Group is a description or investigation of symmetry.

The E8 group deals with a very complex set of symmetries: it represents a 57-dimensional object that has to be described in 248 dimensions. It is mapped as a giant matrix, whose many (205,263,363,600 to be specific) entries are often not numbers, but complex equations in their own right.

David Vogan from MIT explains: "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."

Once the researchers had got this far, they had to refine the model over the course of another year, and it was still too big a problem for a computer to actually run.

The team toyed with the idea of just waiting for technology to catch up, but instead broke the problem down into several smaller versions, each of which would yield an incomplete answer. These could all be stitched back together into a patchwork to recreate the full solution.

The problem, which ran for 77 hours on the Sage supercomputer, yielded a huge amount of data - more than was generated by the human genome project.

According to team leader Jeffrey Adams, the quantity of data is not the only similarity between the two projects: "This is basic research which will have many implications, most of which we don't understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas."

The full implications of the solution might not be realised for many years, but the fact that the problem has been solved has already given hope to another group of researchers: the Atlas project. Their goal is to determine unitary representations of all the Lie Groups - one of the great unsolved mathematical problems of the early 20th century.

The American Institute of Mathematics says the completion of the E8 calculation leaves "little doubt" that their task can be accomplished. ®

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