The Jacobi identity is a touch of associativity: if we deform the usual product of functions in such a way that its associativity is preserved, then we learn at once that, under some natural assumptions, the leading deformation term is a Poisson bracket (on a finite-dimensional real or complex manifold). But what do any directed graphs have to do with this deformation problem? This had not been obvious from H.Weyl--Groenewold--Moyal (1945-46) till the breakthrough result of M.Kontsevich (1997) who proved that on every finite-dimensional affine Poisson manifold, the formal power series (in the deformation parameter h) of the associative non-commutative *-product does exist --- for every choice of the Poisson bracket showing up in the leading deformation term near h^1.
In this talk, we discuss what is known, what is required, and what there remains to do in order to obtain the *-product(s) expansion (those are needed by the physics people from nonlinear or quantum optics, photonics, etc.), how tremendously big the problem is and how fast its size grows (NB: the unknown values of the Riemann zeta function start showing up in the digraph weights at n=7 internal vertices and two sinks, namely zeta(3) does show up), and what our new technology is to constrain the Riemann zeta values by using a desktop computer.
(This is an overview of recent joint work with R.Buring (IM JGU Mainz).)