In the paper Canonical tensor model through data analysis : Dimensions, topologies, and geometries [1], the procedure to calculate the three-way tensor P_{abc} to construct a fuzzy space corresponding to a compact manifold M, was summarized and also the methodology was applied to various fuzzy spaces by using some low dimensional manifolds with various topologies. The main point there was to show the generality of the real symmetric three-way tensors by using those demonstrations and consequently the generality of the Canonical Tensor Model (CTM) in which unlike the tensors in other Euclidean tensor models where the number of ways (the amount of indices) of the tensors are supposed to be equivalent to the dimensions of the building simplicial blocks, the three-way tensors which are real and symmetric can in principle represent any space with free choice of dimensions and topologies.
In this project however, which can be thought of as an extension to the work done in the paper [1], we constructed the fuzzy spaces corresponding to a few more compact manifolds and saw if the idea can be generalized and extended to other topological spaces. In the first half, we constructed the fuzzy space corresponding to the disc after which we moved on to the cone which can be thought of as a slight generalization of the disc after which we constructed the fuzzy space corresponding to the cone with negative curvature which can be thought of as a slight generalization of the cone.
In the second half, we tried to extend the methodology to an orbifold after giving a very brief introduction to an orbifold followed by a few examples. We wanted to study the generality of the real symmetric three-way tensors by extending the methodology to construct the fuzzy spaces corresponding to orbifolds. So we considered the torus and studied the fuzzy space construction corresponding to the manifold first after which we constructed the fuzzy space corresponding to the orbifold obtained from the torus.
We also tried to understand the topological properties of these spaces by using the tools from data analysis. We first used the tensor-rank (or CP) decomposition to decompose a tensor into a number of vectors and then used the other data analysis technique called the persistent homology to understand the various topological properties by regarding the vectors as points forming a space as in [1].