July/August 2007

Simplified Complexity

By Brittany Sauser

On paper, the calculation would cover an area the size of Manhattan. But an international group of 18 mathematicians and computer scientists, including two from MIT, has found a more practical way to calculate the inner workings of E8, one of the most complicated symmetrical structures in mathematics. Mathematics professor David Vogan and Dan Ciubotaru, an instructor in the math department, were among those who put the supercomputer Sage through a 77-hour computation to calculate the 200 billion numbers in E8's character table, generating a 60-gigabyte file.

E8 is an example of a Lie group--a continuous symmetry group whose structure is always transforming yet, like a sphere rotating around an axis, always looks the same. All but five Lie groups fall into one of four classes related to linear algebra and Euclidean geometry. E8 is the most complex of these five "exceptional" Lie groups. It describes the symmetries of a 57-dimensional object that can be rotated in 248 ways without changing its appearance. Calculating its character table--a 453,060-by-453,060 matrix that describes all the ways E8 can appear as a symmetry group--is just one important step toward understanding all Lie groups, says Vogan.

This computer-generated illustration is of the E8 root system, an arrangement of 240 vectors in an eight-dimensional space. The image is a two-dimensional projection of that eight-dimensional arrangement.

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