I will tell a story around a theorem we proved with Tom Hutchcroft and Omer Tamuz. In one version of the story, we look for fixed-point theorems in the spirit of Markov-Kakutani but for several maps at the same time.

In the other version, we cross out items from a finite list and we ask: is there a random list that would almost not change at all when we cross out any of the first few items?

We solve these questions, which are really only one question, by performing what looks like a backwards random walk in which every step would be infinitely long, but stationary.