Scalar Curvature of Causal Graphs

[maxitg]

The project goal

Thanks to @SantiagoGiner from 2021 Winter School, we now have a way of computing CausalDensityDimension (aka Myrheim–Meyer dimension) of causal graphs.

This project will be the next step, which is computing the scalar curvature (aka the Ricci scalar).

Expected deliverables:

• A function or property for computing the scalar curvature given a causal graph.
• A corresponding documentation page that can also be turned into a community post for the summer school.

Motivations

To understand how General Relativity might be related to graph rewriting, we need to understand how to extract geometric properties from graphs. This project will be part of that effort.

External prerequisites

• Familiarity with general relativity or differential geometry.
• Experience with C++ or Wolfram Language.

SetReplace prerequisites

• Understanding of causal graphs.

Difficulty

Given that approaches for doing this already exist in Causal Set Theory, this should not be too difficult to implement.

Potential instructors

@maxitg

Existing work

• CausalDensityDimension (aka Myrheim–Meyer dimension) is already implemented.
• Approaches for computing scalar curvature already exist in Causal Set Theory (see arXiv:1001.2725, also Wiki).

Required future SetReplace features

We need a property for computing causal graphs in Yellowstone (a SetReplace redesign we are currently working on). However, this project can be implemented as a separate function, for which everything is ready.