subscribe: Posts | Comments

A mathematical solution to the excessive dimensions

Norwegian mathematician Sophus Lie discovered the mathematical group called E8, researchers have tried in vain to understand this extraordinarily complex object described by a matrix of numbers over 400 000 rows and columns. But it is now done: an international team of experts using powerful computers and programming techniques has been able to "decode" E8, a feat which is the sequencing of the human genome, which should enable progress in a wide range of problems in geometry, number theory and the physics of string theory.

"Although the sequencing of the human genome is of fundamental importance in biology, it does not provide an immediate cure or a miracle treatment against cancer," notes the mathematician Jeffrey Adams, leader of the project and a professor of mathematics at university Maryland. "Our study is similar: it is a critical basic research, but its implications may not become known for many years."
The "sequencing" E8 is part of a larger project (*) to clarify all Lie groups (which are mathematical descriptions of symmetry for continuous objects like cones, spheres and their counterparts in larger three. Several of these groups are well understood); E8 is the most complex.
The groups … Otherwise
It is easy enough to understand the symmetries of a square, for example. The corresponding group has only two elements: mirror images as diagonals and the mirror images that result from sharing in two as the centre of any of its sides. The symmetries form a group whose members are only those 2 degrees of freedom, or dimensions.The surface of an object as a continuous symmetrical sphere is two-dimensional, because it just only two coordinates (latitude and longitude on Earth) to define a position. But in space, a sphere can be rotated in three axes (x-axis, y-axis and axis "z"), and the group of symmetries corresponding has three dimensions.
In this context, E8 defies the imagination. The symmetries represent a solid 57 dimensions (it takes 57 coordinates to define a position), and the group of symmetries has 248 dimensions.

A collaboration between experts and a machine
Because of its size and complexity, the calculation of E8 has asked about 77 hours a supercomputer Sage and creating a roster of 60 gigabytes. In comparison, the human genome occupies a gigabyte. Task more difficult, the computer must have continuously access to tens of gigabytes of data in its memory (RAM), which is far more capabilities of home computers and even those supercomputers until recently.

The calculations themselves were very sophisticated and need all the science experts in various fields, capable of developing new mathematical techniques and new programming methods.And despite many "crashes" of the computer, for both software and hardware problems, the morning of January 8, 2007 the calculation of E8 ended.

Some figures
The result of the calculation of E8 is a matrix of 453 060 lines on as many columns.There are 205 263 363 600 entries in this matrix, each of which is a polynomial. The largest entry is this:

The value of this polynomial for q = 1 is 60 779 787.
There are 1 181 642 979 separate polynomials in the matrix and 13 721 641 221 coefficients in them. The biggest factor is 11 808 808.
Project funded by the NSF (National Science Foundation) via the IAA (American Institute of Mathematics).

Legend of artwork:
System 240 vectors roots in a space 8 dimensions. These vectors are the summits of an object to 8 dimensions called polytope 421 of Gosset. In 60 years, Peter McMullen was drawn by hand representation in two dimensions. The image shown here is based on the pattern of McMullen and was done on computer by John Stembridge of the University of Michigan.
The straight lines joining the adjacent summits in the polytope, the colors reflect the length of the projection in two dimensions. As this illustration is a two-dimensional projection of an object to 8 dimensions, it represents only a portion of the symmetries of polytope of Gossett.
The algebra Lie E8 east to 248 dimensions: 8 spatial dimensions represented here plus a dimension for each 240 vehicles roots.