Colloquium Mathematics - A.V. Kiselev University of Groningen

Title: Poisson bracket deformations using the Kontsevich graphs: open problems.

Abstract:

We formulate several open problems from the theory of universal --
by using the Kontsevich calculus of oriented graphs -- infinitesimal
symmetries of Poisson brackets on arbitrary finite-dimensional affine
manifolds.
(A). One class of problems concerns the (extremal) combinatorial
properties of those unoriented graph cocycles from which all known
proper nonlinear deformations are built by the orientation morphism.
The other class of open problems is about combinatorial and
topological properties of visibly non-generic oriented graphs that
arise from the corresponding Poisson cocycle factorization equations,
and about the newly discovered topological identities in the spaces of
Leibniz graphs which balance such equations.
   These topics will be interesting to our random graph colleagues.

(B) Thirdly, an open question is whether (or why not) the universal
flows are nontrivial in the Poisson cohomology modulo improper terms
(which vanish by force of the Jacobi identity). Independently, the
action of known symmetries on classes of ``nearly generic'' Poisson
brackets yields a set of open problems from the standard geometry of
PDE and integrable systems.
   This class of open problems will be challenging to experts in
dynamical systems.

   The talk is aimed to provide an overview of these research topics.
   (Joint work with R.Buring.)