Question

Here's a different application. Kontsevich defined for every graph $G$ a hypersurface $Y_G$ in a way motivated by QFT and the theory of Feynmann integrals. Motivated by computer experiments, he suggested that period integrals on the $Y_G$ should always be multiple zeta values. I am not sure of the precise relationship here, but I believe that this is (at least morally) the same thing as stating that the cohomology of $Y_G$ contains only mixed Tate motives. This is a very strong condition to impose and would say that the cohomology of $Y_G$ is extremely special. In particular this would imply that the function $q \mapsto \#Y_G(\mathbf F_q)$ that counts the number of points on $Y_G$ over a finite field is always given by a polynomial in $q$. Belkale and Brosnan in "Matroids, motives and a conjecture of Kontsevich" disproved this conjecture in the strongest possible way: they showed that for ANY scheme $X$ of finite type over $\mathbf Z$, the function $q \mapsto \#Z(\mathbf F_q)$ is a finite linear combination of functions $q \mapsto \#Y_G(\mathbf F_q)$ for graphs $G$. Their proof uses Mnëv's theorem in a crucial way.