**This page contains a summary report on the work
conducted in the framework of the project,
addressed at the general audience.**

The main objective of the project was to contribute towards the development of a theoretical basis and specific applications related to the concept of persistent homology in three domains: digital images, data mining, and dynamical systems.

Topology is an area of mathematics concerned with studying the properties of spaces, such as connectedness or the existence of holes of various types (tunnels, voids, etc.) Algebra is an area of mathematics concerned with studying symbols and equations, and manipulating them. Describing contiguous spaces in a finite way that is suitable for manipulating them with the computer is very demanding if at all possible. However, this is feasible using algebraic topology, a branch of mathematics that uses tools from algebra to study topological spaces. In particular, homology is a quantity that reflects the structure of the space in terms of connected components, holes, and their mutual relation. When a space evolves in time, some features appear and disappear. The persistence of these features during time intervals is described by means of persistent homology. Precisely this notion was the leading motive of the project.

A digital image consists of a collection of rectangular pixels. Images displayed at the computer screen are 2-dimensional and consist of squares. 3-dimensional images describe shapes in the space and consist of cubes. Automatic analysis of such images helps quickly detect specific features and classify the images. Computing persistent topology of structures obtained from such images may be part of such process. Algorithms and methods developed in the framework of the project focus on the computation of advanced algebraic-topological structures (Eilenberg-Zilber contraction and Steenrod squares), and may contribute in the future to the accuracy and reliability of such analysis. Potential applications are immense, and include the detection of cancer cells or other anomalies in biomedical imaging.

Data mining is the automatic analysis of large quantities of data to extract previously unknown interesting patterns. Currently used methods are typically based on statistical analysis, which dates back to Bayes (1700s) and regression analysis (1800s). An approach based upon persistent topology may provide completely new insights into the data, e.g., by determining the structure of the entire dataset. This is a new and dynamically developing interdisciplinary branch of science. In the framework of the project, investigation of possibilities was made towards topological analysis of weakly structured biomedical data (e.g. records of patients), and towards analysis of cardiological data series. Possible future applications include early detection of cardiological problems, as well as computer support for decisions made by medical doctors in choosing the best possible therapy tailored to the individual patient.

A dynamical system is a mathematical concept for describing an object varying in time, using a fixed rule that depends on the current state of the object (and not on its past). Dynamical systems are mathematical models that are used in a multitude of situations to describe a variety of phenomena, such as the growth of a population or spread of an infectious disease. Such models typically depend on several parameters. Mathematical analysis of long-term behavior of the system depending on the parameters is a highly nontrivial task. Algorithmic analysis using topological concepts provides an automatic method for determining robust features of the system, and classifying the various behaviors encountered as the parameters vary in the prescribed ranges. It also provides a wider selection of information than simple numerical simulations or a purely analytical approach. The research conducted within the framework of the project contributed to the development of new methods that broaden the spectrum of dynamical systems to which such automatic analysis can be applied, and also contributed towards involving the concepts of persistent homology in order to improve the efficiency and reliability of such methods. A specific application to a model that describes an infectious disease spreading between two regions connected by transportation was developed, which showed the complexity of the possible dynamics, and thus conveying a warning message to authorities dealing with this kind of issues. The developed methods have a wide spectrum of possible applications, from dynamical systems that model chemical reactions, through animal or plant population growth models, to theoretical and applied physics.

To sum up, the abstract mathematical concept of persistent homology has been proved to be a milestone in algorithmic approach to the analysis of various kinds of objects, from digital images, through large collections of documents or other kinds of data, to models of dynamical phenomena.