El teorema del nervio en el análisis topológico de datos: aplicaciones en problemas de clasificación binaria

In this thesis, we give a brief overview of Topological Data Analysis (TDA) from its math- ematical foundations to state-of-the-art applications. First, we present the necessary con- cepts from Topology, Algebra and Algebraic Topology that make TDA possible. Next, we introduce and prove the Nerve Theorem for convex and compact covers which underlies most of the methods used in TDA. We will also show the relevance of the Nerve Theorem in TDA from the perspective of Category Theory. Then, we explain how persistent homology is defined and used in TDA. Lastly, we present an application published in [1] of the Nerve Theorem and persistent homology that proves a lower bound for the sample size of data points that faithfully recovers the homology of the decision boundary manifold in binary classification problems.