In this project I will work on several topics from convex and integral geometry. The focus will lie
on the study of valuations. These are classical geometric quantities like volume and surface, which
behave naturally with respect to unions and intersections of sets. In the last decades, valuations
were studied in the context of convex bodies, convex functions and on curved objects resp. surfaces
in higher-dimensional spaces, so-called manifolds, and multiple similarities between the different
theories were discovered. The aim of this project is to develop new methods to confirm these
empirical observations and to make the relations explicit. Subsequently, I will apply these methods
in the integral geometry of certain manifolds (complex resp. quaternionic projective spaces) as well
as to solve current problems for valuations on convex bodies resp. convex functions.