Successfully draw maps of one of the most complex mathematical structures

Mathematicians have mapped the inner workings of one of the most complex structures ever studied: this object is the unusual Lie E ₈ structure. This achievement is significant both in terms of basic knowledge progress and because there are many connections between E ₈ and other areas, including string theory and geometry.

The magnitude of this calculation result can be staggering: the calculation results if printed in small sizes, will probably cover the entire Manhattan area. Mathematicians are known to have a working style alone, but the ' E raid' ₈ , part of a large research project, brought 18 mathematicians from the US and Europe together to working closely together for 4 years.

'This is interesting,' said Peter Sarnak, a Princeton professor of mathematics (not involved in the study). 'Understanding and classifying representatives of Lie groups is key to understanding phenomena in various areas of mathematics and science, including algebra, geometry, number theory, and physics. and Chemistry. This work is very valuable for future mathematicians and scientists. '

Larger than the human genome

The experimental system of E ₈ consists of 240 vectors in 8-dimensional space. These vectors are vertices (angles) of 8-dimensional objects called Gosset polytope 421. In the 60s, Peter McMullen drew (by hand) a two-dimensional representation of Gosset polytope 421. The image was created by machine. computer, based on McMullen's drawings. (Photo: Provided by the American Institute of Mathematics)

The magnitude of the E ₈ calculation results has drawn comparisons with the work of human genome research. The human genome, the genome contains all the genetic information of a cell, is less than a gigabyte. E8 calculation results, the result contains all the information about E ₈ and its representatives, the size is 60 gigabytes. So there is enough room to hold songs that can be played continuously for 45 days in MP3 format. While many scientific studies have to deal with a large amount of data, the E8 calculation results are very different: The given data size is relatively small, but the solution itself is very large, and very dense.

Like human genome research, these results are just the beginning. According to Jeffrey Adams, director of research, 'this is the basic research that has many implications, most of these implications are still not understood. Just like the human genome doesn't give you an immediate psychoactive drug, our results are a basic tool that people can use to develop research in other areas. ' This work may not predict the mathematical and physical implications, implications that have not been available for many years.

Mr. Hermann Nicolai, director of the Albert Einstein Institute in Bonn, Germany (not participating in this study) said ' this is an impressive achievement. While mathematicians have long known about the beauty and uniqueness of the E8 structure, we, physicists, must appreciate its special role just recently - though However, with our effort to integrate gravity with other fundamental forces into proper quantum gravity theory, we now have to confront it in almost every corner! Therefore, understanding the inner workings of the E8 structure is not only great progress for mere mathematicians, but can also help physicists in finding a unified theory. '

Results of calculation of E8 structure

The team that calculated the structure of E ₈ started the study four years ago. They meet at the American Institute of Mathematics every summer, and work together in small groups throughout the year. Their work requires mathematical theory and complex computer programming.

The team member, Mr. David Vogan from MIT said, 'The material for this subject is very much and very difficult to understand. Even after we understood the basic math, it took us over 2 years to do it on the computer. '

And then comes the problem of finding a computer big enough to do calculations. It took another year for the team to make the calculations more efficient so that it could fit into the current supercomputers, but it still exceeded the capacity of these computers' hard drives.

The team had to expect to wait for a larger computer, so Noam Elkies of Harvard University found a way to implement some small versions of this calculation, each of which would produce a session. Incomplete version of the answer. These incomplete answers can be put together to give the final answer. The cost is to do this calculation up to 4 times, plus time to combine incomplete answers together. Finally, it takes about 77 hours to perform calculations on the Sage supercomputer.

This graph describes the mathematical structure similar to the E8 structure but is much smaller than E8.(Photo: David Vogan, MIT)

Beautiful symmetry

At the basic level, the calculation of E8 structure is the study of symmetry . Mathematicians invented the Lie group to grasp the nature of symmetry: underlining any symmetrical object, like a sphere, is a Lie group.

Lie groups are divided into groups. Classical groups such as A ₁ , A ₂ , A ₃ , . B ₁ , B ₂ , B ₃ , . C ₁ , C ₂ , C ₃ , . and D ₁ , D ₂ and D ₃ , . climbed like hills running gently toward the horizon. Protruding from this mathematical landscape are jagged peaks of special groups G ₂ , F ₄ , E ₆ , E ₇ , which surpass all of them as E ₈ . E ₈ is the most unusual complex group : It is the symmetry of a separate 57-dimensional object and the E ₈ structure itself is 248-dimensional.

To describe this new result requires a further level of abstraction. The ways that E ₈ represents itself as a symmetry group are called representations. The goal is to describe all possible representations of E ₈ These representations are extremely complex, but mathematicians have described them in the form of basic unified blocks. The new result is a complete list of these unified blocks for representatives of E ₈ , and an accurate description of the relationships between them, all decoded in a matrix with 205,263,363,600 entries. .

Atlas of Lie group research works

The calculation of structure E ₈ is part of an ambitious research known as 'The Atlas of Lie Group Studies and Representatives' . The goal of this atlas study is to identify the unique representations of all Lie groups. This is one of the biggest unsolved problems since the beginning of the 20th century. The success of calculating E ₈ structure has alleviated many doubts about the possibility that the atlas group could be completed. This research work.

The research atlas is funded by the National Science Foundation through the American Institute of Mathematics.

Thanh Van