This project aims to (i) resolve challenging graph problems in distributed and dynamic settings, with a focus on connectivity problems (such as computing edge connectivity and distances), and (ii) on the way develop a systematic approach to attack problems in these settings, by thoroughly exploring relevant algorithmic and complexity-theoretic landscapes. Tasks include<br/><br/><br/>- building a hierarchy of intermediate computational models so that designing algorithms and proving lower bounds can be done in several intermediate steps, <br/><br/>- explaining the limits of algorithms by proving conditional lower bounds based on old and new reasonable conjectures, and<br/><br/>- connecting techniques in the two settings to generate new insights that are unlikely to emerge from the isolated viewpoint of a single field.<br/><br/><br/>The project will take advantage from and contribute to the developments in many young fields in theoretical computer science, such as fine-grained complexity and sublinear algorithms. Resolving one of the connectivity problems will already be a groundbreaking result. However, given the approach, it is likely that one breakthrough will lead to many others.