E8, The Theory of Everything

   

   Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E₈. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry. The magnitude of the calculation is staggering: the answer, if written out in tiny print, would cover an area the size of Manhattan. Mathematicians are known for their solitary work style, but the assault on E8 is part of a large project bringing together 18 mathematicians from the U.S. and Europe for an intensive four-year collaboration. 

"It's a mathematical entity that we know exists, but we had to explore its inner structure," says Hermann Nicolai, a mathematical physicist at the Albert Einstein Institute in Potsdam, Germany, who was not involved in the work. "It's a bit like making a plan of a complicated building, or exploring an ancient pyramid to see how it was built."

   Mathematicians take these descriptions to wild extremes by imagining the 3D objects in myriad dimensions. The group E8 encapsulates the symmetries of a geometric object like a sphere, cylinder or cone, but in 57 dimensions. E8 itself has 248 dimensions. Check out the visual representation E8 in a video after the break.

   The team (called Atlas) 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. 

via: NewScientist & AIMath