Digital Persistent homology

 Persistent homology is a mathematical technique used in topological data analysis to study the shape and structure of data. In the context of discrete metric values representing a complex system, persistent homology can be applied to analyze the topological features of the system across different spatial scales. Here's an equation representing the application of persistent homology to discrete metric values:

Let $�$ be a set of data points representing a discrete metric space. The discrete metric values between pairs of points in $�$ can be represented as a matrix $�$, where ${�}_{��}$ represents the distance between points $�$ and $�$ in the discrete metric space.

1. Construction of Simplicial Complex:

   Using the discrete metric values, construct a simplicial complex $�$ by forming a simplex for every subset of points whose pairwise distances are below a certain threshold $�$. This threshold represents the spatial scale at which we are analyzing the topology of the system.

   Let ${�}_{�}$ represent the simplicial complex constructed at scale $�$.

2. Computation of Homology Groups:

   Compute the homology groups of the simplicial complex ${�}_{�}$ using algebraic topology techniques. This involves finding cycles and boundaries within the simplicial complex, leading to the computation of Betti numbers ${�}_{�}$, which represent the $�$th Betti number of the complex.

3. Persistence Diagram:

   Represent the persistence information as a persistence diagram. In the persistence diagram, each point $(�,�)$ represents the birth ($�$) and death ($�$) scales of a topological feature (e.g., connected component, tunnel, void) in the simplicial complex. Features with longer lifespans are more persistent across different spatial scales.

4. Persistent Homology Equation:

   The persistent homology equation involves analyzing the persistence diagram and extracting topological features that persist across different scales. The overall equation for persistent homology can be written as follows:

   $$\text{PH}=\{({�}_1,{�}_1),({�}_2,{�}_2),\dots ,({�}_{�},{�}_{�})\}$$

   Where $({�}_{�},{�}_{�})$ represents the persistence point corresponding to the $�$th topological feature in the persistence diagram.

In this equation, persistent homology provides a way to understand the topological features that exist in the discrete metric space $�$ across various spatial scales. By analyzing the persistence diagram, researchers can identify significant topological features and gain insights into the underlying structure of the system.

Applying persistent homology to the Discrete Curvature Tensor from Triangulation and the Triangulated Einstein-Hilbert Action involves analyzing the topological features of a discrete spacetime model at various spatial scales. Here's how you can formulate the application of persistent homology in this context:

1. Discrete Curvature Tensor from Triangulation:

   Given a triangulated discrete spacetime, the Discrete Curvature Tensor represents the discrete analogue of the curvature of spacetime. This tensor captures the geometric properties of the discrete spacetime at each simplex. Let ${�}_{��}$ be the Discrete Curvature Tensor for the $�$th and $�$th simplices in the triangulated spacetime.

2. Triangulated Einstein-Hilbert Action:

   The Triangulated Einstein-Hilbert Action describes the dynamics of the discrete spacetime model. It incorporates the Discrete Curvature Tensor and other geometric quantities, representing the discrete counterpart of Einstein's equations in general relativity.

   $${�}_{��}=\sum _{\text{simplices}}{�}_{��}\cdot \text{other geometric quantities}$$

   Here, ${�}_{��}$ is the Einstein-Hilbert Action for the triangulated spacetime.

3. Persistent Homology Analysis:

   • Construction of Simplicial Complex:

     Utilize the simplices from the triangulated spacetime to construct a simplicial complex $�$. The distance metric between simplices can be defined using the Discrete Curvature Tensor values.

   • Persistence Diagram:

     Compute the persistent homology of the simplicial complex $�$ at different spatial scales. Analyze the persistence diagram to identify topological features that persist across various scales.

4. Incorporating Triangulated Einstein-Hilbert Action:

   Integrate the Triangulated Einstein-Hilbert Action into the persistent homology analysis. Consider the action as a weighting factor when constructing the simplicial complex. This weighting reflects the influence of the action on the persistence of topological features.

   $$\text{Weighted Distance}=\text{Distance metric}\times \text{Action term}$$

   Modify the construction of the simplicial complex and compute persistent homology with the weighted distances.

5. Analysis and Interpretation:

   Analyze the persistence diagram considering the weighted distances. Persistent features represent the topological structures that are stable under the influence of both geometry (captured by the Discrete Curvature Tensor) and dynamics (described by the Triangulated Einstein-Hilbert Action).

   • Identify significant topological features that persist across various spatial scales.
   • Explore the relationship between the action term and the persistence of these features.
   • Draw conclusions about the interplay between geometry, dynamics, and the resulting spacetime topology in the discrete model.

This application of persistent homology provides a method to understand how the geometry and dynamics of a discretized spacetime influence its topological characteristics at different scales, offering insights into the complex interconnections between geometry, topology, and the underlying physics of the system.