Differentialgeometrisk Analyse. Kurver. Flader. Mangfoldigheder.

Steen Markvorsen (SM) forsker i geometrisk analyse og differentialgeometri med særlig vægt på betydningen af krumningsbegrebet for kurver, flader, og for generelle mangfoldigheder, herunder f.eks. krumningens grundlæggende lokale og globale indflydelse på geodætiske afstands-kurver, varmeledning, Laplace spektret, isoperimetri og topologi.

• Geometry and differential geometry with special emphasis on diffusion related processes in doubly curved surfaces - for example in surfaces with constant mean curvature (soap films and foams). We obtain sharp inequalities for mean exit times for Brownian motion and for isoperimetry (the area to length ratio) for special spherical subdomains on such surfaces. The methods are analytic (potential theory and heat equation analysis) and geometric (curvature comparison geometry). There are applications to biological membranes resembling minimal surfaces and to surfaces with complex but curvature controlled topology. This research is supported by the Danish Natural Sciences Research Council (FNU) and by the Spanish Ministry of Science via collaboration with Vicente Palmer, and Ana Hurtado, Universitat Jaume I.
  
• Geodesic curves are ocally the shortest curves on a given surface. In curved surfaces and in topologically non-trivial surfaces we encounter, however, an interesting lensing effect: At a suitable distance from a given point we may find other points from which there are more than one geodesic curve to the given point. There are applications of this research to models of forest fires and to the study of gravitational lenses.
  

Dissemination activities and interests

• Matematicum, the Mathematics Inspiratorium at The Technical University of Denmark. This initiative is supported by FNU, by the Birch & Krogboe Foundation, by Danske Bank, and by Rector's strategic fund at DTU.
  
• Didactically motivated applications of Computer Algebra and Analysis Systems (CAAS) for disseminating and unfolding structural and abstract mathematical concepts and ideas. Has been supported by LearningLab DTU.