In this work we present the concepts of simplicial complexes, homology, filtrations, persistent homology, persistent modules and barcodes. On one hand we introduce a notion of distance between persistence modules, algebraic objects which describe persistent homology groups. On the other hand we introduce a notion of distance between barcodes, which are a way of visualising of persistent modules. Using these concepts we state and prove the algebraic stability theorem for q-tame persistent modules. This theorem is the key argument which confirms that persistent homology is a useful tool when dealing with noisy data.
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