The Mesoscopic category, Automata and Tropical Geometry

We start with comparisons of hierarchies in Biology and relate it to Quan- tum Field Theories. Thereby we discover many similarities and translate them into rich mathematical correspondences. The basic connection goes via scale transformations and Tropical geometry. One of our core observations is that Cellular Automata can be naturally introduced in many physical models and refer to a (generalised) mesoscopic scale, as we point it out for the fundamental relation with Gauge Field Theories. To illustrate our framework further, we apply it to the Witten-Kontsevich model and the Miller-Morita-Mumford classes, which in our picture arise from a dynamical system.