(Joint work with Ben Knudsen): Reinterpreting the seminal paper of Smale that launched our field, we shall show that among robots that have access to randomness, there is a price to be paid for being "digital" as opposed to being "analog". Besides analog complexity, one also has "analog category" - which has an Eilenberg-Ganea theorem, but has surprising properties for finite groups!

(Similar notions were introduced also by Dranishnikov and Jauhari with different interpretations and motivations.)

The considerations are similar to ones involved in computing "universal" parametrised complexity for certain manifolds. Depending on what happens during the talk, we might discuss some of this direction (which is joint also with Nick Wawrykow).