Science News & Science Current Events
 
Email a Friend Send to a friend
Printer Friendly Print Researchers master one of the largest, most complicated mathematical structures

Researchers master one of the largest, most complicated mathematical structures

March 21, 2007

New 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



Related Mathematical Structure Current Events and Mathematical Structure News Articles
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?
More Mathematical Structure Current Events and Mathematical Structure News Articles


Mathematical Structures for Computer Science
by Judith L. Gersting



Discrete Mathematical Structures (6th Edition)
by Bernard Kolman, Robert Busby, Sharon C. Ross

Key Message: Discrete Mathematical Structures, Sixth Edition, offers a clear and concise presentation of the fundamental concepts of discrete mathematics. This introductory book contains more genuine computer science applications than any other text in the field, and will be especially helpful for readers interested in computer science. This book is written at an appropriate level for a wide...



The Structure of Economics: A Mathematical Analysis
by Eugene Silberberg

This text combines mathematical economics with microeconomic theory and can be required or recommended as part of a course in graduate microeconomic theory, advanced undergraduate or graduate-level mathematical economics, or any advanced topics course. It also has reference value for international, library, professional and reference markets. This revision addresses significant new topics that...

Mathematical Structures of Language (Tracts in Pure & Applied Mathematics)
by Zellig S. Harris

Mathematical Structures for Computer Science
by Judith L. Gersting



Discrete Mathematical Structures: Theory and Applications
by D.S. Malik, M.K. Sen

Discrete Mathematical Structures teaches students the mathematical foundations of computer science, including logic, Boolean algebra, basic graph theory, finite state machines, grammars, and algorithms. Authors Malik and Sen employ a classroom-tested, student-focused approach that is conducive to effective learning. Each chapter motivates students through the use of real-world, concrete examples,...

Structure of Language & Its Mathematical
by Roman Jakobson



Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics (Cambridge Mathematical Library)
by David Ruelle

Reissued in the Cambridge Mathematical Library, this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. Background material on physics has been collected in appendices to help the reader. Supplementary work is provided in the form of exercises and problems that were "open"...



Theory of Mathematical Structures
by Jirí Adámek



Modern Algebra and the Rise of Mathematical Structures
by Leo Corry

The book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-nineteenth century to its consolidation by 1930, and then it considers several attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea. ...

© 2008 BrightSurf.com