|At the department of Mathematics, the Geometry group is mainly working with the development and the applications of geometric and topological properties of curves and surfaces as well as their generalizations to manifolds. For this we apply methods from analysis, topology and discrete mathematics and the applications are within areas such as computer geometry, quantum geometry, and protein geometry. We work at all scales from the subatomic quantum scale to the geometry of space. Through tools from classical geometry it is now possible to disclose the properties of e.g. nanotubes or biological structures within the cell. Applications of geometric and topological methods in physical and biological systems are used in the design of curves and surfaces with specific properties of enormous value to industry.
Geometric insight and research
There are a number of clear and fundamental concepts and results – such as the concept of a manifold, the classification of space forms, and Morse theory, to mention but a few – which in essence pass on deep knowledge and insight from the development in the twentieth century concerning geometry, differential geometry and global analysis and which at the same time carry great potentials for the future. We expect the topics of geometric invariants and comparison geometry to be of continuing interest for the group as well as the connection between Riemannian geometry and discrete geometry.
Metric and topological descriptors in combination with local curvature invariants are effective tools for the classification and comparison of geometric structures which possess well defined measures of distance. There are applications of such results within the analysis of docking properties of large bio-molecules and the classification of proteins. Part of our research is concerned with the ‘translations’ of methods and results in between discrete geometry and Riemannian geometry.
Computer experiments naturally play an essential and increasing role in this work. The figures and the Moineau pump case from Grundfos represent but two examples of experimental differential geometry, where we use the computer both as a bridge and as a vehicle to move effectively both ways between the “concrete” and the “abstract”.
Curvature geometry and Laplace geometry
The secrets of the curvature tensor and its deep – but still in many ways open – potential for describing discrete as well as differentiable geometries are being explored with the purpose of finding and applying effective geometric invariants. For example, one part of our research is concerned with the influence of curvature on naturally occurring processes on – and deformations of – submanifolds (membranes, proteins, minimal surfaces, etc.) in well defined ambient spaces. The processes under consideration are e.g. heat transport and solutions to the Schrödinger equation.
The Laplace operator induces a well defined Fourier analysis on Riemannian manifolds and indeed on quite general metric spaces. There is a wealth of results and applications awaiting further research within this area between – and yet combining – differential geometry, analysis and discrete mathematics. The results so far are mainly concerned with surfaces in 3-space but it seems that the category of spaces as well as the volume of applications can be enlarged considerably.
Flight of cross-shaped balsawood boomerangs. The theory of flight for a boomerang can be explained mathematically.