Computational topological methods are increasingly and successfully being used to analyze and understand complex data, provided both from experimental sources and numerical simulations. Many of these topological methods use homology theory and the related Betti numbers to concentrate on essential topological information. Usually the objects of interest are discretized before their homology can be computed. This process of discretization can introduce artifacts which cause the resulting topological information to be incorrect. Uncertainty can also occur in experimental settings due to noise. In this project, the investigator and his colleagues develop two mechanisms for a quantitative assessment of the correctness of topological information. The first approach is based on the realization that most applied problems involve randomness, and it studies the accuracy of homology computations for random fields by providing explicit probability estimates which give a-priori information on suitable problem discretizations. The second mechanism develops a method for rigorous computational a-posteriori correctness based on randomized subdivisions. In addition, the investigators address two specific applications which are interesting in their own right. First, they develop a new method for finding isolating blocks in dynamical systems, which has a wide range of possible applications and provides a novel tool for studying dynamical properties of evolution equations. The second application focuses on pattern classification in materials science and leads to a classification of possible phase separation phenomena in multi-component alloys.
Recent advances in computational sciences have led to an immense increase in the amount of generated data, and the development of novel techniques to quickly and reliably extract essential information from this data has become a central issue. The project research develops probabilistic approaches for topological data analysis, including quantitative reliability assessments. The project has applications in the field of materials science, such as in the study of the properties of industrial level metal alloys and the design of new materials. Topological methods have already been shown to provide important insight into such materials studies, and the techniques developed in this project are applied in the context of materials characterization and design. This leads to a classification of phase separation phenomena in multi-component alloys and has the potential to lead to system-response-maps, which can then be used in the design of materials, such as for example the design of controlled drug-release coatings for arterial stents. The project also leads to a broader study of random fields using homology theory, which in the long term has applications in medical imaging. The project involves student research at both the undergraduate and graduate levels.