[info] [tt] Math research team maps E8

[Eugen Leitl]

http://web.mit.edu/newsoffice/2007/e8.html

Math research team maps E8 Calculation on paper would cover Manhattan

March 18, 2007

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.

For more information on E8 visit http://aimath.org/E8/.


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