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SUMMARY:Asymptotic higher ergodic invariants of magnetic lines - Akhmet'ev
 \, P (IZMIRAN)
DTSTART:20121207T094000Z
DTEND:20121207T102000Z
UID:TALK41925@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:V.I.Arnol'd in 1984 formulated the following problem: "To tran
 sform 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 divergence-free fields by magnetic f
 ields. Asymptotic invariants of magnetic fields\, in particular\, the theo
 rem by V.I.Arnol'd about asymptotic Gaussian linking number\, is a bridge\
 , which relates differential equitations and topology. We consider 3D case
 \, the most important for applications. \n\nAsymptotic invariants are deri
 ved from a finite-type invariant of links\, which has to be satisfied corr
 esponding limit relations. Ergodicity of such an invariant means that this
  invariant is well-defined as the mean value of an integrable function\, w
 hich is defined on the finite-type configuration space $K$\, associated wi
 th magnetic lines. \n\nAt the previous step of the construction we introdu
 ce a simplest infinite family of invariants: asymptotic linking coefficien
 ts. The definition of the invariants is simple: the helicity density is a 
 well-defined function on the space $K$\, the coefficients are well-defined
  as the corresponding integral momentum of this function. Using this gener
 al construction\, a higher asymptotic ergodic invariant is well-defined. A
 ssuming the the magnetic field is represented by a $delta$-support with co
 ntains 3 closed magnetic lines equipped with unite magnetic flows\, this h
 igher invariant is equal to the corresponding Vassiliev's invariant of cla
 ssical links of the order 7\, and this invariant is not a function of the 
 pairwise linking numbers of components. When the length of generic magneti
 c lines tends to $infty$\, the asymptotic of the invariant is equal to 12\
 , this is less then twice order $14$ of the invariant.  \n\nPreliminary re
 sults arXiv:1105.5876 was presented at the Conference "`Entanglement and L
 inking"' (Pisa) 18-19 May (2011). \n
LOCATION:Seminar Room 1\, Newton Institute
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