Local constraint solving - how to colour without looking (much)
How can individuals cooperate to satisfy local constraints without a central authority? Examples might include autonomous drones navigating in a swarm, or university departments scheduling their seminars. Individuals can make random choices and communicate with each other, but all must follow the same procedure.
How small can we make the "coding radius" - the distance to which an individual must communicate? In the setting of the integer line Z, there is a surprising universal answer that applies to every non-trivial constraint problem. In d-dimensional Euclidean space, answers are available for the key case of proper colouring; it turns out that there is a huge difference between 3 and 4 colors. Finally, I'll mention how changing the question slightly has led to the discovery of an amazing mathematical object that seemingly has no right to exist.