We revisit rough paths and signatures from a geometric and "smooth model" perspective. This provides a lean framework to understand and formulate key concepts of the theory, including recent insights on higher-order translation, also known as renormalization of rough paths. This first part is joint work with C Bellingeri (TU Berlin), and S Paycha (U Potsdam). In a second part, we take a semimartingale perspective and more specifically analyze the structure of expected signatures when written in exponential form. Following Bonnier-Oberhauser (2020), we call the resulting objects signature cumulants. These can be described - and recursively computed - in a way that can be seen as unification of previously unrelated pieces of mathematics, including Magnus (1954), Lyons-Ni (2015), Gatheral and coworkers (2017 onwards) and Lacoin-Rhodes-Vargas (2019). This is joint work with P Hager and N Tapia.
Peter K Friz is Einstein Professor in Mathematics at TU-Berlin, part of the Research Group: Probability Theory and Mathematical Finance. He is also affiliated to the Weierstrass Institute for Applied Analysis and Stochastics.