This dissertation focuses on developing theoretical tools in the field of Topological Data Analysis and more specifically, in the study of toroidal dynamical systems. We make contributions to the development of persistent homology by proving Kunneth-type theorems, to topological time series analysis by further developing the theory of sliding window embeddings, and to multiscale data coordinatization in topological spaces by proving stability theorems. First, in classical algebraic topology, the Kunneth theorem relates the homology of two topological spaces with that of their product. We prove Kunneth theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. Using these theorems, we also develop novel methods for algorithmic and abstract computations of persistent homology. One of the direct applications of these results is the abstract computation of Rips persistent homology of the N-dimensional torus.Next, we develop the general theory of sliding window embeddings of quasiperiodic functions and their persistent homology. We show that the sliding window embeddings of quasiperiodic functions, under appropriate choices of the embedding dimension and time delay, are dense in higher dimensional tori. We also explicitly provide methods to choose these parameters. Furthermore, we prove lower bounds on Rips persistent homology of these embeddings. Using one of the persistent Kunneth formulae, we provide an alternate algorithm to compute the Rips persistent homology of the sliding window embedding, which outperforms the traditional methods of landmark sampling in both accuracy and time. We also apply our theory to music, where using sliding windows and persistent homology, we characterize dissonant sounds as quasiperiodic in nature.Finally, we prove stability results for sparse multiscale circular coordinates. These coordinates on a data set were first created to aid non-linear dimensionality reduction analysis. The algorithm identifies a significant integer persistent cohomology class in the Rips filtration on a landmark set and solves a linear least squares optimization problem to construct a circled valued function on the data set. However, these coordinates depend on the choice of the landmarks. We show that these coordinates are stable under Wasserstein noise on the landmark set.
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