My primary research interests lie in the area of stochastic processes that take values in function spaces, and in particular in the development of theory and methodology for function-valued time series with time-dependent characteristics. This is a line of research which I started to develop during my PhD, and is concerned with the analysis of sequential collections of data points that themselves come in the form of complex mathematical structures, such as curves, surfaces or manifolds. Inference techniques to analyze such data not only require a mathematically rigorous and quantitative study of their `shape' and dependence structure, but moreover must translate into computationally efficient methods. Examples can be found in (neuro-)imaging, climatology, genomics, and econometrics. I am especially interested in the development of appropriate statistical theory to further advance inference methods in these essential applications, which are characterized by dependence over time and space, and of which the dependence structure is of an evolutionary nature.