The main theme of the project is persistent homology in three contexts: digital images, data mining, and dynamical systems. Persistent homology is the most important innovation that has yet emerged from the young field of computational topology. It finds various applications, and in each of them provides new qualities and novel methods.Effective and reliable methods for the analysis of digital images are highly demanded, especially with the increasing technological capabilities of capturing multiple high-resolution images e.g. in medicine. Data mining techniques for acquiring knowledge from huge collections of data, including text documents, are bound to be a new scientific methodology of the future. Dynamical systems appear ubiquitously in modeling of population models, chemical reactions and other processes, and automatic analysis of qualitative properties of the dynamics is of great importance for the understanding of the model.The main objectives of the project are: to use persistent homology for automatic determination of an optimal thresholding level and denoising method for the analysis of digital images; to optimize the existing methods and to develop new algorithms for the persistent homology approach to the analysis of large collections of data, with emphasis on text documents; and to develop new methods and algorithms for applying the persistent homology approach to the analysis of discrete-time dynamical systems induced by continuous maps. In addition to the theoretical basis and algorithms for each of the domains, effective software aimed at specific applications will also be developed.The project involves various branches of mathematics and computer science, from algebraic topology to graph theory, and has trans-disciplinary nature with a wide potential of applications, particularly in natural sciences, and thus is expected to contribute to strengthening national and international scientific collaboration.