A useful idea to study an object is, it turns out, not to just consider the object but an entire space of objects like it. Consider, for instance, the situation in which you want to simplify an object to one you understand (like in the solution of the Poincaré conjecture) or when the space is naturally evolving over time (like spacetime): in this case, it is often easier to understand the space of spaces, rather than the space itself, as difficulties like singularities can vanish once you have taken a broader, bird’s eye view.
I want to discuss a specific case of this, based on a great talk Daria Poliakova gave in our seminar. If you don’t like it, well…
For a cup of coffee more 