Topological methods in dynamics and data analysis
Topological data analysis (TDA) is a new practical method in applied mathematics that analyses data that are high-dimensional, incomplete and noisy using techniques from topology. Persistent homology is a key technical tool in TDA that identifies clusters, holes and voids in topological structures. Research conducted within the EU-funded project PHIDM (Persistent homology - Images, data and maps) led to the development of new methods to further study topological features of data. Automatic analysis of 3D images enables quick detection of specific features and classification of images. This can greatly benefit biomedical imaging for early detection of cancer cells or other anomalies. New algorithms and methods developed in PHIDM focused on the computation of advanced topological structures and may enhance accuracy and reliability of images' automatic analysis. Data mining is a computational process of discovering unknown interesting patterns in large data sets. Currently used methods are typically based on statistical analysis methods that date back to 1700. Refining information from persistent homology is a relatively new interdisciplinary field that helps obtain good qualitative understanding of the entire data set. Scientists investigated the possibility of using persistent homology for topological analysis of weakly structured biomedical data (e.g. records of patients) and of cardiological data series. Possible future applications include early detection of cardiological problems, as well as computer-assisted decision support programmes for diagnosis and treatment. Dynamical systems are mathematical models describing a variety of phenomena, such as the growth of a population or spread of an infectious disease. Algorithmic analysis using topological concepts provides an automatic method for determining robust features and classifying behaviours. Research conducted within PHIDM led to broadening the spectrum of dynamical systems to which automatic analysis can be applied. Persistent homology applied to a model describing spread of an infectious disease between two regions connected by transportation infrastructure was developed, showing the complex dynamics. The newly developed methods have a wide spectrum of possible applications, ranging from dynamical systems that model chemical reactions through animal or plant population growth models, to theoretical and applied physics.
Persistent homology, image analysis, data mining, topological data analysis, dynamical systems