From cracks to catastrophes, “singularity theory” could shed light

[Roger Bagula]

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.”