Geometric curvature functionals: energy landscape and discrete methods

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This research proposal focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces, or more general k-dimensional sets in Euclidean d-space. Previous investigations of the principal investigators concentrated on the regularizing effects of such energies, with a priori estimates that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. The present project aims at a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Their impact on geometric knot theory is going to be investigated, and suitable structure-preserving discrete versions need to be developed and analysed. The tools range from non-local fractional differential operators, measure theory and integral geometry, to variational calculus and geometric topology. This project touches related mathematical disciplines such as knot theory, harmonic analysis, discrete differential geometry, and Riemannian geometry, and there are interesting connections to the Sciences and Engineering.