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Researchers master one of the largest, most complicated mathematical structures
March 21, 2007New tool could drive breakthroughs in several disciplines
Ever since 1887, when Norwegian mathematician Sophus Lie discovered the mathematical group called E8, researchers have been trying to understand the extraordinarily complex object described by a numerical matrix of more than 400,000 rows and columns.
Now, an international team of experts using powerful computers and programming techniques has mapped E8—a feat numerically akin to the mapping of the human genome—allowing for breakthroughs in a wide range of problems in geometry, number theory and the physics of string theory.
"Although mapping the human genome was of fundamental importance in biology, it doesn't instantly give you a miracle drug or a cure for cancer" said mathematician Jeffrey Adams, project leader and mathematics professor at the University of Maryland. "This research is similar: it is critical basic research, but its implications may not become known for many years."
Team member David Vogan, a professor of mathematics at the Massachusetts Institute of Technology (MIT), presented the findings today at MIT.
The effort to map E8 is part of a larger project to map out all of the Lie groups—mathematical descriptions of symmetry for continuous objects like cones, spheres and their higher-dimensional counterparts. Many of the groups are well understood; E8 is the most complex.
The project is funded by the National Science Foundation (NSF) through the American Institute of Mathematics.
It is fairly easy to understand the symmetry of a square, for example. The group has only two components, the mirror images across the diagonals and the mirror images that result when the square is cut in half midway through any of its sides. The symmetries form a group with only those 2 degrees of freedom, or dimensions, as members.
A continuous symmetrical object like a sphere is 2-dimensional on its surface, for it takes only two coordinates (latitude and longitude on the Earth) to define a location. But in space, it can be rotated about three axes (an x-axis, y-axis and z-axis), so the symmetry group has three dimensions.
In that context, E8 strains the imagination. The symmetries represent a 57-dimensional solid (it would take 57 coordinates to define a location), and the group of symmetries has a whopping 248 dimensions.
Because of its size and complexity, the E8 calculation ultimately took about 77 hours on the supercomputer Sage and created a file 60 gigabytes in size. For comparison, the human genome is less than a gigabyte in size. In fact, if written out on paper in a small font, the E8 answer would cover an area the size of Manhattan.
While even consumer hard drives can store that much data, the computer had to have continuous access to tens of gigabytes of data in its random access memory (the RAM in a personal computer), something far beyond that of home computers and unavailable in any computer until recently.
The computation was sophisticated and demanded experts with a range of experiences who could develop both new mathematical techniques and new programming methods. Yet despite numerous computer crashes, both for hardware and software problems, at 9 a.m. on Jan. 8, 2007, the calculation of E8 was complete.
National Science Foundation
"Although mapping the human genome was of fundamental importance in biology, it doesn't instantly give you a miracle drug or a cure for cancer" said mathematician Jeffrey Adams, project leader and mathematics professor at the University of Maryland. "This research is similar: it is critical basic research, but its implications may not become known for many years."
Team member David Vogan, a professor of mathematics at the Massachusetts Institute of Technology (MIT), presented the findings today at MIT.
The effort to map E8 is part of a larger project to map out all of the Lie groups—mathematical descriptions of symmetry for continuous objects like cones, spheres and their higher-dimensional counterparts. Many of the groups are well understood; E8 is the most complex.
The project is funded by the National Science Foundation (NSF) through the American Institute of Mathematics.
It is fairly easy to understand the symmetry of a square, for example. The group has only two components, the mirror images across the diagonals and the mirror images that result when the square is cut in half midway through any of its sides. The symmetries form a group with only those 2 degrees of freedom, or dimensions, as members.
A continuous symmetrical object like a sphere is 2-dimensional on its surface, for it takes only two coordinates (latitude and longitude on the Earth) to define a location. But in space, it can be rotated about three axes (an x-axis, y-axis and z-axis), so the symmetry group has three dimensions.
In that context, E8 strains the imagination. The symmetries represent a 57-dimensional solid (it would take 57 coordinates to define a location), and the group of symmetries has a whopping 248 dimensions.
Because of its size and complexity, the E8 calculation ultimately took about 77 hours on the supercomputer Sage and created a file 60 gigabytes in size. For comparison, the human genome is less than a gigabyte in size. In fact, if written out on paper in a small font, the E8 answer would cover an area the size of Manhattan.
While even consumer hard drives can store that much data, the computer had to have continuous access to tens of gigabytes of data in its random access memory (the RAM in a personal computer), something far beyond that of home computers and unavailable in any computer until recently.
The computation was sophisticated and demanded experts with a range of experiences who could develop both new mathematical techniques and new programming methods. Yet despite numerous computer crashes, both for hardware and software problems, at 9 a.m. on Jan. 8, 2007, the calculation of E8 was complete.
National Science Foundation
Geometry shapes sound of music
Through the ages, the sound of music in myriad incarnations has captivated human beings and made them sing along, and as scholars have suspected for centuries, the mysterious force that shapes the melodies that catch the ear and lead the voice is none other than math.
Harvard scientists predict the future of the past tense
Verbs evolve and homogenize at a rate inversely proportional to their prevalence in the English language, according to a formula developed by Harvard University mathematicians who've invoked evolutionary principles to study our language over the past 1,200 years, from "Beowulf" to "Canterbury Tales" to "Harry Potter."
Sandia researchers develop contaminant warning program for EPA to monitor water systems in real time
Sandia National Laboratories researchers are working with the U.S. Environmental Protection Agency (EPA), University of Cincinnati and Argonne National Laboratory to develop contaminant warning systems that can monitor municipal water systems to determine quickly when and where a contamination occurs.
What can a magnet tell you about rain patterns? More than you would guess
If someone said you can understand rain patterns and the dynamics of the atmosphere by studying magnets and magnetism — and therefore make better predictions of the effects of global warming — would you think he's crazy? Brilliant?
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