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Mathematicians Map E8

March 18, 2007, Source: The American Institute of Mathematics


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.

"This is exciting," said Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton University (not affiliated with the project). "Understanding and classifying the representations of Lie Groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, Physics and Chemistry. This project will be valuable for future mathematicians and scientists."
 

Bigger than the Human Genome

The magnitude of the E8 calculation invites comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. That is enough space to store 45 days of continuous music in MP3 format. While many scientific projects involve processing large amounts of data, the E8 calculation is very different: the size of the input is comparatively small, but the answer itself is enormous, and very dense.
Like the Human Genome Project, these results are just the beginning. According to project leader Jeffrey Adams, "This is basic research which will have many implications, most of which we don't understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas." This could have unforeseen implications in mathematics and physics which do not appear for years.

According to Hermann Nicolai, a director of the Max Planck Institute in Potsdam, Germany (not affiliated with the project), "This is an impressive achievement. While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently --- yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner! Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory."
 

The E8 Calculation


The team 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.
 

Beautiful Symmetry

At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.
Lie groups come in families. The classical groups A1, A2, A3, ... B1, B2, B3, ... C1, C2, C3, ... and D1, D2, D3, ... rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional!

To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries.
 

The Atlas of Lie Groups Project


The E8 calculation is part of an ambitious project known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations of all the Lie groups. This is one of the great unsolved problems of mathematics, dating from the early 20th century. The success of the E8 calculation leaves little doubt that the Atlas team will complete their task.

The Atlas team consists of about 20 researchers from the United States and Europe. The core group consists of Jeffrey Adams (University of Maryland), Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah) , Marc van Leeuwen (Poitiers), David Vogan (MIT), and (until his death in 2006) Fokko du Cloux (Lyon).

The results of the E8 computation were announced in a talk at MIT by David Vogan on Monday, March 19, 2007.

The Atlas project is funded by the National Science Foundation through the American Institute of Mathematics.
 

'Lie group E8' math puzzle solved after 120 years

Calculation on paper would cover Manhattan
March 18, 2007. Source: MIT press release

Math research team maps E8

An international team of 18 mathematicians, including two from MIT, has mapped one of the largest and most complicated structures in mathematics. If written out on paper, the calculation describing this structure, known as E8, would cover an area the size of Manhattan.

The work is important because it could lead to new discoveries in mathematics, physics and other fields. In addition, the innovative large-scale computing that was key to the work likely spells the future for how longstanding math problems will be solved in the 21st century.

MIT's David Vogan, a professor in the Department of Mathematics and member of the research team, will present the work today, Monday, March 19 at 2 p.m. in Room 1-190. His talk, "The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness," is open to the public.

E8, (pronounced "E eight") is an example of a Lie (pronounced "Lee") group. Lie groups were invented by the 19th-century Norwegian mathematician Sophus Lie to study symmetry. Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects.

Mathematicians study symmetries in higher dimensions. E8 has 248 dimensions.

"What's attractive about studying E8 is that it's as complicated as symmetry can get. Mathematics can almost always offer another example that's harder than the one you're looking at now, but for Lie groups E8 is the hardest one," Vogan said.

"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Jeffrey Adams, project leader and a mathematics professor at the University of Maryland. "This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large-scale computing to solve complicated mathematical problems."

The mapping of E8 may well have unforeseen implications in mathematics and physics that won't be evident for years to come.

"There are lots of ways that E8 appears in abstract mathematics, and it's going to be fun to try to find interpretations of our work in some of those appearances," said Vogan. "The uniqueness of E8 makes me hope that it should have a role to play in theoretical physics as well. So far the work in that direction is pretty speculative, but I'll stay hopeful."

"This is an exciting breakthrough," said Peter Sarnak, a professor of mathematics at Princeton University and chair of the scientific board at the American Institute of Mathematics (AIM). "Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists," said Sarnak, who was not involved in the work.

The magnitude and nature of the E8 calculation invite comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes. This is enough to store 45 days of continuous music in MP3-format.

The mapping of E8 is also unusual because it involved a large team of mathematicians, who are typically known for their solitary style. "People will look back on this project as a significant landmark and because of this breakthrough, mathematics will now be viewed as a team sport," said Brian Conrey, executive director of AIM.

The E8 calculation is part of an ambitious project sponsored by AIM and the National Science Foundation known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations--roughly speaking, symmetries of a quantum mechanical system--of all the Lie groups (E8 is the largest of the exceptional Lie groups). This is one of the most important unsolved problems of mathematics. The E8 calculation is a major step and suggests that the Atlas team is well on the way to solving this problem.

The Atlas team consists of 18 researchers from around the globe. The core group consists of Adams and Vogan, plus Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah), Marc van Leeuwen (University of Poitiers) and (until his death in 2006) Fokko du Cloux (University of Lyon). Additional team members include Dan Ciubotaru, the CLE Moore Instructor in MIT's Department of Mathematics, and Alfred Noel, a professor at the University of Massachusetts at Boston and an MIT visiting scholar.
 

Western Michigan University (WMU) mathematician helps crack E8 puzzle

March 23, 2007. Source: WMU News

KALAMAZOO--A Western Michigan University mathematician is part of an international team of researchers that recently solved one of the toughest problems in mathematics and received international media attention for its work.

After four years of intensive collaboration, WMU's Dr. Annegret Paul, associate professor of mathematics, and 17 other mathematicians and computer scientists successfully mapped a 120-year-old puzzle. The team, with creative minds hailing from the United States and Europe, was convened by the American Institute of Mathematics in Palo Alto, Calif., to map a theoretical object known as "Lie group E8." The Atlas of Lie Groups Project is funded by the National Science Foundation.

Lie (pronounced Lee) groups were invented by the 19th century Norwegian mathematician Sophus Lie to express the symmetry of three-dimensional objects such as spheres, cones and cylinders. E8 is the most sophisticated Lie group with 248 dimensions, and it was long considered impossible to solve.

The final result of the E8 calculation was unveiled March 19 at the Massachusetts Institute of Technology, during a news conference in Boston Paul attended. Partners included MIT, Cornell University, University of Michigan, University of Utah and University of Maryland.

The team's work is continuing as members prepare to tackle similar mathematical problems that could yield an infinite number of calculations.

Mapping E8 is a "stepping stone," according to Paul, who came to WMU in 1999. It is one of the most symmetrical mathematical structures in the universe, and symmetry can provide critical insights into a problem. The mapping of E8 could lead to discoveries in mathematics, physics and other fields and new technology.

"The breakthrough is being able to translate these mathematical questions into something that a computer can do," Paul says. "There is still al lot more we need to do."

E8 is so complicated that its handwritten solution would cover a grid that would measure more than seven miles on each side, so large that it could cover a piece of paper the size of Manhattan. It is the most complicated group, but not the longest, Paul says.

To understand using E8 and all its possibilities requires calculation of 200 billion numbers. The problem's proof involves about 60 times as much data as the Human Genome Project, which contains all the genetic information of a cell.

Media contact: Deanne Molinari, (269) 387-8400, deanne.molinari@wmich.edu
 

International team solves 248-dimensional maths puzzle

March 20, 2007. Source: Cordis News, Data Source Provider: American Institute of Mathematics (AIM)


A team of European and US scientists has mapped one of the most complicated structures ever studied: the exceptional Lie group E8. This could have huge implications for understanding of algebra, geometry, number theory, quantum gravity and chemistry.

Lie groups lie at the intersection of two fundamental fields of mathematics: algebra and geometry. They are named after Norwegian mathematician Sophus Lie, who studied them at the end of the 19th century.

Working together in the four-year Atlas project are 18 mathematicians from France, the US and Canada.

According to the American Institute of Mathematics, 'At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.'

Classical groups are described as resembling 'gentle rolling hills towards the horizon'. More complicated groups are described as 'jagged peaks', and towering above them all is E8, 'an extraordinarily complicated group'. E8 represents the symmetries of a particular 57-dimensional object, and is itself 248-dimensional.

What the team has succeeded in doing is describing each of the building blocks for E8, as well as the relations between them. The matrix has 205,263,363,600 entries, and if written out in tiny print, would be large enough to cover an area the size of Manhattan.

A comparison with the mapping of the human genome gives another clear indication of the sheer size of the matrix. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation is 60 gigabytes in size.

As with the Human Genome Project, the full implications of the mapping will not be known for many years. 'This is basic research which will have many implications, most of which we don't understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas,' said project leader Jeffrey Adams.

Hermann Nicolai, Director of the Albert Einstein Institute in Bonn, Germany, explained the importance of the achievement for physics. Physicists have come across E8 much more recently than mathematicians, but encounter it regularly in attempts to unify gravity with other fundamental forces into a consistent theory of quantum gravity. 'Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory.'
 

 

 

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Last updated: March 21, 2007