The mathematical mapping E8

A research team of mathematicians from the Massachusetts Institute of Technology has mapped one of the most complicated mathematical structures ever defined

The subject was named: "E8 exceptional Lie group" after the Norwegian mathematician Lee Sophos/ because of the links of E8 to other fields of science, this is a significant achievement for basic knowledge, including string theory in geometry.

The amount of calculations is amazing, and if it were written on paper in small print it would cover all of Manhattan. The mathematicians who developed this work are known for a unique but rigorous work style. This is part of a wider project known as "Mathematics 18" developed in cooperation between mathematicians from the USA and Europe intensively for 4 years. Eugene Higgins Eugene Higgins - professor of mathematics at Princeton University - who is not a partner in the development of the project. "Understanding and ranking the importance of presenting this model is critical to directing various phenomena in many fields of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be of great value to mathematicians and scientists in the future.

larger than the human genome

From the calculation rate of E8, the comparison of development with the cracking of the human genome is requested. The size of the human genome, which contains all the generic information of the cell, is less than a GigaByte. The result of E8 calculation which contains all the information represented in E8 is at the rate of GB60. Such an amount is enough to store continuous music in MP3 format for 45 days.

While many scientific research projects contain a large amount of data, E8's calculation is completely different: the amount of input data is relatively small but the result of the calculation is huge in size and very compressed.

As with the project of cracking the human genome, these results are only the beginning of the road. According to the words of the head of the project, Jeffrey Adams, "This basic research has many implications, most of which we do not yet understand. Just as the human genome does not provide an immediate panacea, neither do our results, which are a basic tool that will be used for advanced research in other fields."

This may have unexpected consequences in mathematics and physics that have not been observed for many years.

According to the words of Hermann Nicolai, director of the Max Planck Institute in Patsdam - Germany (who is not a partner in the project), "This is an impressive achievement. While mathematicians have long been aware of the beauty and uniqueness of E8, we physicists have only recently come to understand the concept's extraordinary role in our efforts to unify gravity with other fundamental forces in quantum theory gravity, we now have access to almost every corner! Thus, understanding the work of E8 in depth, is not only an advancement of pure mathematics, but will also help physicists in their search for a theory of unified power."
E8 calculation

The team that developed the calculation of E8 started its work four years ago. They meet each summer at the American Institute of Mathematics and smaller working groups meet annually. Their work requires a combination of theoretical mathematics and complex computer programming.

According to the words of a member of the team David Vogan from MIT: "The literature on this subject is dense and very difficult to understand. Even after we understood the mathematical basis of the idea, it took us another two years to implement it on a computer. Then began the problem of finding a computer powerful enough to perform the calculation. Another year was required to optimize the calculation so that it would fit into an existing supercomputer capable of performing the calculations, but there still remained the problem of the worsening capacity that makes it possible to contain the data of the calculation results."

The team considered waiting for a larger computer to arrive, when Noam Atkins of Harvard University discovered an ingenious way to make slight changes to the calculation that each show a partial version of the result. These partial results add up to the complete solution. The cost of this solution was in the time required to perform the calculations - four times the time required for the original calculation, plus the additional time required to compile the results. All in all, it takes 77 hours on the supercomputer.

The "beautiful" symmetry

At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie group to find a way to define symmetry for any symmetric object. For example, a symmetric body like a sphere in the three-dimensional space is a Li bundle. Li bundles come in families.

The classical groups A1, A2, A3, … B1, B2, B3, … C1, C2, C3, … D1, D2, D3, … rise as rolling hills towards the horizon. Standing out above the mathematical landscape, pushed to the summits of the extraordinary groups G2, F4, E6, E7 and above them all is E8. E8 is a particularly complicated bunch: it contains symmetric objects with 57 dimensions and it itself has 248 dimensions!

An additional level of abstraction is required to describe the results. The way in which E8 declares itself as symmetry is called "representations". The goal is to describe all the possible presentations of E8. These presentations are extremely complicated, but the mathematicians describe them in terms of basic block structures, and the resulting result is a list of these structures for presentations of E8 and a precise description of the relationships between them, all encoded in a matrix with 205,263,363,600 entries.

The atlas of the Lee Group project

The calculation is part of a prestigious project known as: "Atlas of Lie Groups and Representations"

The aim of the Atlas project is to uniquely define each presentation of the Lee group. This is one of the great, unsolved problems of mathematics presented at the beginning of the twentieth century. The success of the E8 calculation minimizes the doubt that the Atlas Project team will complete its task.

Translated by Yoram Tomer from American Institute of Mathematics press release About a discussion held at the MIT Institute under the direction of David Vogan on Monday 19.3.2007: "E8 calculation results"