Quantization of linear Poisson structures and degrees of maps
  Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain the geometric meaning of this series as degrees of maps of some grand configuration spaces; the associativity proof is also interpreted in purely homological terms. An interpretation in terms of degrees of maps shows that any other 1-form on the circle also leads to a *-product and allows one to compare these products. 
Michael Polyak