Graph Rewriting for Lattice Topological Invariants

We are going to study finite lattice systems in the context of simple rewriting rules: based on discrete manifolds and Cayley graphs, in this project we look for the set and taxonomy of simple rewriting rules that preserve certain properties that can later be understood as conserved measurable magnitudes of a physical system, like the Brouwer degree or winding number. By investigating those toy models about field configurations of discrete fiber bundles, we will later apply local perturbations to a field configuration, looking for discrete analogs to Noether's theorem. Furthermore, via coloured graphs, with this approach we will aim to characterise (energetically stable) physical-inspired systems in the context of many-particle quantum finite distributions in highly symmetric arrays or low-dimensional spin-like states as domain wall in magnetization theory, being local solutions of minimal information configurations. Even mesoscopic scale is responsible for exotic topological defects; with this discretization, we will try to understand the computational first-principle mechanism that allows nontrivial winding numbers and the material and geometries to host pseudo-particle structures like skyrmions, bubbles or merons. We think that this inductive reasoning here would offer a valuable approach for finding new phenomena or even reinterpreting old ones.