Mathematics, Computer Science

Herbert Edelsbrunner is one of the worlds leading researchers in computational geometry and topology. This subdiscipline of computer science and mathematics concerns the development of algorithms to solve geometrical and topological problems and their application in various branches of engineering and the natural sciences. Edelsbrunner made a significant contribution to the development of computational geometry in the last quarter of the 20th century and went on to become one of the founders of computational topology around the turn of the millennium. This new area can be regarded as the logical extension of computational geometry, but it is based on different mathematical foundations, which explains why only a few researchers work in both fields. Currently, Edelsbrunner is pursuing three main research areas, each with the goal of further developing computational topology and thus opening up new areas of application. These emphases include stochastic, algebraic and geometric questions within the field of topological data analysis. The ambitious goal of the first stochastic research area is the detailed analysis and description of expected persistence diagrams. This approach unifies a large number of questions in probability theory and suggests others. The aspiration of the second algebraic research area is the efficient and effective calculation of the persistence of multidimensional, filtered spaces, which promises a wide range of applications. The complexity of the situation is such, however, that the most that can be hoped for are partial successes. The third geometric research area concerns grids in space and packing properties of balls. All three aspects are necessary to gain the possibly revolutionary insights needed to understand the microscopic world of materials. Drawing on the Wittgenstein Award funding, Edelsbrunner will be able to develop Vienna and Austria into a leading international research hub of computational geometry and topology. With the Wittgenstein funds, research goals can be achieved more quickly, and previously unexplored applications can be approached using topological methods. In some cases, alternative solutions will arise, with advantages and disadvantages as compared to conventional methods. In others, however, new research may open the door to undreamed- of possibilities.

The research conducted under this project addresses topological, discrete, stochastic, and computational aspects of geometric questions motivated by applications in the sciences, and primarily but not exclusively in the biological and medical sciences. Here are a few highlights: - The expected number of edges, triangles, and higher-dimensional simplices in the Delaunay mosaic of a stationary Poisson point process is studied through the lens of discrete Morse theory. This way the precise expressions---which were known up to three dimensions since the breakthrough work of Roger Miles around 1970---have been extended to four dimensions. Furthermore, we generalize the results to the weighted case and to higher-order Delaunay mosaics. - The concept of persistent homology is extended to the chromatic case, which is anticipated to have major applications in material science and in cancer biology. This includes the topological development of six related persistence diagrams, the discrete analysis of the underlying geometric structures, and fast algorithms and software to construct them. - The maximum Betti numbers of a union of finitely many unit spheres (or rather spherical unit balls) in three dimensions have been known to be between linear and quadratic in the number of spheres. We establish that it is indeed quadratic, but the persistence of all but a linear number of these holes is less than a constant. Importantly, the different aspects of the geometric questions are related, and an orchestrated multi-pronged approach made progress possible on all intended fronts.