A primer on Script Geometry

TL;DR

Script Geometry introduces a novel approach to discrete differential geometry that extends beyond simplicial complexes, emphasizing minimality and local homology to develop a comprehensive framework for discrete function theory.

Contribution

It presents the foundational concepts, properties, and operations of Script Geometry, including metrics, duals, and operators, expanding discrete differential geometry beyond traditional simplicial complexes.

Findings

Introduces the concept of tightness as a minimality condition.

Establishes the equivalence of a Poincare lemma in Script Geometry.

Provides concrete examples illustrating the framework.

Abstract

This text is an exposition of a new approach into discrete differential geometry, called Script Geometry. In difference to classic approaches while scripts are based on complexes of cells we are not limited to simplicial complexes. One of the principal concepts of Script Geometry is the notion of tightness which is a minimality condition corresponds to the condition that the local homology at the level of cells is trivial and provides us the equivalence of a Poincare lemma. Basic concepts and properties of scripts are presented as well as operations on scripts. Also the necessary concepts of metrics, dual scripts, Dirac and Laplace operators are presented to provide the groundwork for a discrete function theory. To help understanding the concepts concrete and illustrative examples are provided.