Geometrie

Abstract

The official program consisted of 20 lectures and therefore left plenty of space for fruitful informal collaboration for the 44 participants. One emphasis with 9 talks at this meeting was on geometric flows. Exciting progress could be reported on

• existence results for mean curvature as well as for the Ricci flow with singular initial data,
• stability and convergence results for the Ricci flow and new invariant curvature conditions,
• an extension of Perelman's work to open 3-manifolds with positive scalar curvature, and an improved singularity analysis.

Another important theme was metric and Finsler geometry with five talks covering the following topics

• a survey on currents in metric spaces, and existence of a curvature tensor in a measured sense on Alexandrov spaces which are noncollapsed limits,
• existence of path isometries to the Euclidean space for a wide range of metric spaces,
• closed geodesics on Finsler manifolds and volume entropy of Hilbert's Finsler metrics on convex sets.

The other six talks covered other aspects of geometry including

• dynamics of topological holomorphic maps on 2-sphere and geodesics of the Weil–Petersson metric,
• a cohomogeneity one example of a positively curved manifold and Einstein–Ricci-soliton solvmanifolds,
• a discrete analogue of conformal equivalence and isoparametric hypersurfaces.

Several connections between the different areas became apparent during the workshop. For example, the initial value problem for the Ricci flow with singular initial data is closely linked to smoothing problems occuring in Alexandrov geometry.