  This entry contains the core material of my habilitation thesis, soon to be
officially submitted. It provides a self-contained presentation of the original
results in this thesis, in addition to their detailed proofs. The motivation of
these results is the analysis of data which lie in Riemannian manifolds. Their
aim is to bring about general, meaningful, and applicable tools, which can be
used to model, and to learn from such "Riemannian data", as well as to analyse
the various algorithms which may be required in this kind of pursuit (for
sampling, optimisation, stochastic approximation, ...). The world of Riemannian
data and algorithms can be quite different from its Euclidean counterpart, and
this difference is the source of mathematical problems, addressed in my thesis.
The first chapter provides some taylor-made geometric constructions, to be used
in the thesis, while subsequent chapters (there are four more of them), address
a series of issues, which arise from unresolved challenges, in the recent
literature. A one-page guide, on how to read the thesis, is to be found right
after the table of contents.
