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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