Concept of Persistent Homology, it contains below sub-concepts:

Simplicial Complexes: A simplicial complex is a collection of simplexes (points, line segments, triangles, and their higher-dimensional counterparts) that are used to construct a topological space. Filtrations: A filtration is a nested sequence of simplicial complexes, typically indexed by a parameter, such as time or scale. As the parameter changes, new simplexes are added, creating a growing sequence of complexes. This allows us to observe the evolution of topological features over different scales. Homology: Homology is a mathematical concept that captures the topological features of a space in different dimensions. For example: ( H_0 ) captures connected components (0-dimensional features). ( H_1 ) captures loops (1-dimensional features). ( H_2 ) captures voids or cavities (2-dimensional features), and so on. Computing homology gives a summary of the “holes” in different dimensions. Persistence: In the context of persistent homology, persistence refers to the lifespan of topological features as we vary the scale parameter. Features that appear and disappear quickly (short-lived) are often considered noise, while those that persist over a wide range of scales (long-lived) are considered significant. Barcode and Persistence Diagram: Barcode: A visual representation where each topological feature is represented by a horizontal line segment. The x-axis typically represents the scale parameter, and the length of the line segment represents the persistence of a feature. Persistence Diagram: A plot where each feature is represented by a point in a 2D plane, with the birth time on the x-axis and the death time on the y-axis. Significant features will lie far from the diagonal line (where birth equals death).

the visual below is powerful illustration. it’s by Bei Wang at Utah University. Hence persistent homology has wide usage including senser network (Reeb graph is a conceptual tool used in topological data analysis and computational geometry to study the shape and features of a topological space. It captures the evolution of connected components of level sets (subsets of the data that share a common value of a function) as the function value changes. This is particularly useful for understanding and visualizing the structure of data in higher dimensions.), gene analysis etc.

Concept of genius genus. My primitive grasp of it is that it’s somewhat like holes. But mathematically, genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.

Genus is closely related to another important topological invariant called the Euler characteristic ( \chi ), which is given for a surface by the formula: [ \chi = V – E – F ]
where ( V ) is the number of vertices, ( E ) the number of edges, and ( F ) the number of faces in a given triangulation of the surface.

genus of a sphere is 0, a donut/torus is 1 and a double torus’s genus number is 2. But confusion arises on sphere, there is an empty void enclosed by sphere surface, why genus is 0? because the discussion is about 2D surface, not 3D, and also to think about holes as if some rubber band that can NOT be shrunk to a point.

when extending the notion of topological invariants to 3-dimensional spaces, we do encounter concepts analogous to genus but adapted for higher dimensions. In three-dimensional topology, a variety of other invariants and properties come into play to describe the structure and complexity of 3D manifolds.

• Betti Numbers: These generalize the notion of counting holes in higher dimensions.
  • ( b_0 ): The number of connected components.
  • ( b_1 ): The number of one-dimensional holes (loops or tunnels).
  • ( b_2 ): The number of two-dimensional holes (voids or cavities).
  • And so on for higher dimensions.

For example Consider the 3-dimensional torus, ( T^3 ):

( b_3 ) is the maximal 3-dimensional “hole” (the whole volume inside ( T^3 )).

Fundamental Group: ( \pi_1(T^3) = \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} ), representing the three independent loops.

Betti Numbers: ( b_0 = 1 ), ( b_1 = 3 ), ( b_2 = 3 ), ( b_3 = 1 ).

( b_0 ) indicates one connected component.

( b_1 ) counts three independent loops.

( b_2 ) counts three independent “voids” or 2-dimensional surfaces wrapping around like a doughnut.