Topological data analysis and applications

Topological data analysis is interested in problems relating to nonlinear systems, large scale data and development of more accurate models, that contribute to a high level research. In the study of time-series data we can identify problems that focus aspects of that nature, with applications to digital disease detection, gene expression, viral evolution, etc. In particular, it describes the qualitative aspects of the data, focusing the topological features with longest lifetimes. To encode the latter, it uses novel mathematical methods that can provide diagrams which are a clear and practical tool that allow us the detection of outliers and to capture the dynamics of the system. In this paper we review this methodology and discuss several applications that can lead to new perspectives to modelling and simulation in system behaviour.