Roger Bagula
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 Posted: Mon Jun 09, 2008 9:20 pm Post subject: From cracks to catastrophes, “singularity t heory” could she |
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http://www.world-science.net/othernews/080604_singularity.htm
From cracks to catastrophes, “singularity theory” could shed light
June 5, 2008
Courtesy European Science Foundation
and World Science staff
There’s often more to everyday events than meets the eye. The
folding of paper, or dripping of water from a tap, are two
examples: they both involve the creation of points known as
singularities.
Singularities occur at places of cutoff or of sudden change, as
in formation of cracks, lightning strikes, creation of ink drops in
printers, and the breaking of a cup when it drops. These points
require sophisticated mathematical techniques to describe,
analyse and predict.
Lightning is one phenomenon exhibiting singularities. Others
include crumpled paper and dripping water. (Image courtesy NASA)
Scientists say many singularities have much in common at all size
scales—from microscopic interactions to the formation of the
universe itself during the so-called Big Bang. But these seemingly
disparate events are usually studied by different scientists in
relative isolation.
A workshop organised by the European Science Foundation in
Paris in January was one of the first attempts to unify the field
of singularities by bringing together experts in the different
fields from astronomy to nanoscience—the study of atomic-scale
structures.
The meeting was aimed at developing common mathematical
approaches to singularities. Improved understanding of the
underlying math would have many benefits, for example in making
materials more resistant to breaking, researchers say.
The event was a success and and paved the way for further research
with greater cross-pollination of ideas, said the convenor, Jens
Eggers of the foundation.
The workshop confirmed, scientists said, that most or all
singularities, from microscopic cracks to the Big Bang, share a
key property known as self-similarity. This means that under
magnification the event looks almost the same. For example a
crack in a piece of plastic exhibits the same jagged structure when
magnified, say, 100 times. This means common mathematical
approaches can be applied.
But the devil is in the details when it comes to comparing
different types of singularities, workshop participants
cautioned. Different systems might have some common features such
as self-similarity, but also unique aspects that require
specialised study. One aim of the workshop was to identify the
common methods that could be applied as a foundation for more
detailed specific study.
Jay Fineberg of Hebrew University in Jerusalem, for example,
presented investigations of cracks in structures or rock
formations. Fineberg discussed new experiments involving gels,
allowing the crack’s structure to be determined in great detail
down to microscopic dimensions, yielding some unexpected findings.
Cracks are often surprisingly complex, Eggers noted, with “many
small side branches, which appear to have complicated, if not
fractal, structure.” Fractal structure here means much the same as
self-similarity, involving a geometric pattern that looks
unchanged under magnification or reduction.
Another example concerned the singularities of crumpling in
paper, presented by Tom Witten of the James Franck Institute in
Chicago. Crumpled paper comprises many ridges and tips that defy
simple analysis. There are many unanswered questions even in
describing each individual cone-shaped tip, Eggers said;
figuring out the underlying math would not just help understand
what happens when we crumple paper, but also other physical
systems involving ridges and tips, such as the way biological
molecules fold into their characteristic forms.
One branch of singularity theory is “catastrophe theory,” which
rose to prominence in the 1970s, initially developed by French
mathematic René Thom and expanded by U.K. mathematic Erik
Zeeman. Catastrophe theory deals with events with space-and-time
components, such as collisions between wave fronts, Eggers said.
“In that case, a problem that takes place in all of space can be
reduced to a problem that takes place along certain lines,” known as
caustics, “which can be classified according to catastrophe
theory.” But not all singularity problems are amenable to this
simplification.
The subject “cuts across disciplines and specializations, such as
experimental physics, theoretical physics, and rigorous
mathematical proofs,” Eggers said. “This workshop very much
reflected this fact, as we had speakers from very different
backgrounds.” |
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