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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Asymptotic higher ergodic invariants of magnetic l
ines - Akhmet'ev\, P (IZMIRAN)
DTSTART;TZID=Europe/London:20121207T094000
DTEND;TZID=Europe/London:20121207T102000
UID:TALK41925AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/41925
DESCRIPTION:V.I.Arnol'd in 1984 formulated the following probl
em: "To transform asymptotic ergodic definition of
Hopf invariant of a divergence-free vector field
to Novikov's theory\, which generalizes Withehead
product in homotopy groups"'. \n\nWe shall call di
vergence-free fields by magnetic fields. Asymptoti
c invariants of magnetic fields\, in particular\,
the theorem by V.I.Arnol'd about asymptotic Gaussi
an linking number\, is a bridge\, which relates di
fferential equitations and topology. We consider 3
D case\, the most important for applications. \n\n
Asymptotic invariants are derived from a finite-ty
pe invariant of links\, which has to be satisfied
corresponding limit relations. Ergodicity of such
an invariant means that this invariant is well-def
ined as the mean value of an integrable function\,
which is defined on the finite-type configuration
space $K$\, associated with magnetic lines. \n\nA
t the previous step of the construction we introdu
ce a simplest infinite family of invariants: asymp
totic linking coefficients. The definition of the
invariants is simple: the helicity density is a we
ll-defined function on the space $K$\, the coeffic
ients are well-defined as the corresponding integr
al momentum of this function. Using this general c
onstruction\, a higher asymptotic ergodic invarian
t is well-defined. Assuming the the magnetic field
is represented by a $delta$-support with contains
3 closed magnetic lines equipped with unite magne
tic flows\, this higher invariant is equal to the
corresponding Vassiliev's invariant of classical l
inks of the order 7\, and this invariant is not a
function of the pairwise linking numbers of compon
ents. When the length of generic magnetic lines te
nds to $infty$\, the asymptotic of the invariant i
s equal to 12\, this is less then twice order $14$
of the invariant. \n\nPreliminary results arXiv:
1105.5876 was presented at the Conference "`Entang
lement and Linking"' (Pisa) 18-19 May (2011). \n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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