Topological Applications of p-Adic Divergence and Gradient Operators

TL;DR

This paper introduces p-adic divergence and gradient operators leading to new Laplacian operators, enabling the analysis of Turing patterns, graph topology, and Mumford curves through heat kernel methods.

Contribution

It develops novel p-adic differential operators and demonstrates their application in topological graph analysis and algebraic geometry.

Findings

Euler characteristic expressed via heat kernel traces

Heat kernels used to extract topological info from Mumford curves

New p-adic operators connect topology and spectral analysis

Abstract

$p$-Adic divergence and gradient operators are constructed giving rise to $p$-adic vertex Laplacian operators used by Z\'u\~niga in order to study Turing patterns on graphs, as well as their edge Laplacian counterparts. It is shown that the Euler characteristic of a finite graph can be expressed via traces of certain heat kernels associated with these new operators. This result is applied to the extraction of topological information from Mumford curves via heat kernels.