Geometry past and present - a stroll through curved spaces

Martina Zähle (Jena)

Abstract:

We offer an excursion through geometry and analysis of the last 150 years, focusing on some complete systems 
of Euclidean invariants - the continuous motion invariant valuations. Convex geometry, differential geometry, 
geometric measure theory and algebraic geometry provide different approaches to these functionals and their 
measure theoretic counterparts on various classes of sets. In special situations they are known, e.g., as 
quermassintegrals or as Lipschitz-Killing curvatures and (lower-dimensional) volumes, including the topological 
Euler number. We  also discuss how to describe the geometry of self-similar sets by means of current fractal versions.