We will discuss special hypersufaces coming from Feynman integrals, their cohomology, class in the Grothendieck ring of varieties and the number of rational points over finite fields. I will mention the Kontsevich conjecture and a counterexample related to a K3 surface. I will explain the independence of the $c_2$ invariant from the representation space of the Feynman period.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/5312