The Mesoscopic category, Automata and Tropical Geometry

TL;DR

This paper explores the connections between biological hierarchies, quantum field theories, and tropical geometry, highlighting how cellular automata relate to mesoscopic scales and gauge theories, with applications to mathematical physics models.

Contribution

It introduces a novel framework linking cellular automata, tropical geometry, and physical models, providing new insights into mesoscopic scales and their mathematical representations.

Findings

Cellular automata can be integrated into physical models at mesoscopic scales.

Tropical geometry serves as a bridge between biological hierarchies and quantum field theories.

Application to the Witten-Kontsevich model demonstrates the framework's relevance.

Abstract

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.