|COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring.|
Asymptotic higher ergodic invariants of magnetic lines
If you have a question about this talk, please contact Mustapha Amrani.
Topological Dynamics in the Physical and Biological Sciences
V.I.Arnol’d in 1984 formulated the following problem: “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”’.
We shall call divergence-free fields by magnetic fields. Asymptotic invariants of magnetic fields, in particular, the theorem 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.
Asymptotic invariants are derived from a finite-type invariant of links, which has to be satisfied corresponding limit relations. Ergodicity of such an invariant means that this invariant is well-defined as the mean value of an integrable function, which is defined on the finite-type configuration space $K$, associated with magnetic lines.
At the previous step of the construction we introduce a simplest infinite family of invariants: asymptotic linking coefficients. 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 general construction, a higher asymptotic ergodic invariant is well-defined. Assuming the the magnetic field is represented by a $delta$-support with contains 3 closed magnetic lines equipped with unite magnetic flows, this higher invariant is equal to the corresponding Vassiliev’s invariant of classical links of the order 7, and this invariant is not a function of the pairwise linking numbers of components. When the length of generic magnetic lines tends to $infty$, the asymptotic of the invariant is equal to 12, this is less then twice order $14$ of the invariant.
Preliminary results arXiv:1105.5876 was presented at the Conference ”`Entanglement and Linking”’ (Pisa) 18-19 May (2011).
This talk is part of the Isaac Newton Institute Seminar Series series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
Other listsC.P. Snow Lectures Cambridge Cancer Centre seminars Quantitative cell biology symposium: June 18 2009
Other talksALICE, PINOCCHIO, FANTASY, AND INTERNATIONAL STEREOTYPES Searching for hidden moving targets on graphs Spectral Sequences Applied to Two Specific Problems in the BRST Cohomology of Supersymmetric Theories in D=4 and D=10 Spacetime dimensions. CEZANNE IN HIS TIME The influence of infant feeding and disease morbidity on children's growth: evidence from the London Foundling Hospital, 1893-1919 The genomics of adaptation in plants: moving from model to non-model species