Metric geometry and Finsler structures of low regularity

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Finlser geometry has been an important subject in differential geometry throughout the 20th. century. It has recently attracted a strong renewed interest with new methods, problems and directions being developed as well as interesting connections with other parts of mathematics. However a number of important examples of Finsler manifolds do not satisfy these smoothness or convexity conditions and many of the standard methods break down. Our project aims to close this gap. In the most extreme setup we merely assume the manifold to be Lipschitz, and the Finsler metrics to be a measurable, possibly degenerate function. In this research project we principally aim at a deeper understanding of the geometry and analytic properties of non smooth and/or non strongly convex Finsler manifolds and their role in geometry. geometric analysis and applied mathematics. The techniques draw alot from metric geometry and therefore our research program contain some topics which formally belong to metric geometry. More specifically we shall investigate the following topics:1. Develop the theory of rectifiable curves in weak metric spaces.2. Study isometric embedding of weak metric spaces to weak Banach spaces.3. Clarify the relation between weak metrics on a manifold that are Lipschitz and weak Finsler structures.4. Describe weak metric spaces admitting a non trivial dilation.5. Study the geodesics and a class of ``special geodesic'' on weak Finsler spaces.6. Investigate the various natural Laplacians