A new metaphor for my research:
I try to learn things about topology by studying objects called "formal groups". What formal groups are doesn't matter so much as the landscape they fill out -- a twisted, mountainous one where the ground clusters up into ever-higher peaks. We can map the lowlands very well, and we have pretty good maps right around the peaks, but it's generally hard to change from the lowland map to a peak map as you start climbing a mountain. However, it turns out you can map the thin rings of foothills around the mountain in a way that *partially* matches up with a map of the lowlands. As the rings get thinner, the matching gets better and better, until in the limit you get: a perfect map of nowhere.
And a thought about the research process:
I went to the perfectoid spaces conference to get insight from number theorists on this foothills mapping problem, and strictly speaking they weren't much help. I was probably asking the wrong questions -- really, most of the questions I have are probably wrong to them, as the objects I care about are somehow simpler -- just groups acting on rings! But without being there, I wouldn't have the idea I have now. It's an idea about complicated things, perfectoid spaces and so on, that turns out to translate well into the simple language of groups acting on rings. I feel, for once, vindicated in the acquisition of broad knowledge that I do almost compulsively.