I've realised a couple of times over the break that when you're able to set aside a block of time for free research (even just e.g. a full afternoon), the benefits of having that unconstrained time are super nice in some quite specific ways.
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To give an example, with a long or open-ended slot, it become much easier to warm up on a problem, and try a few calculations. Moreover, you can even have the time to try a "low-percentage" calculation, which you don't think has a great chance of working out, but you still think is worth trying.
I'm not sure that I'd prioritise such calculations in cases where I e.g. have half an hour before teaching next, for a mix of reasons, chiefly i) there's lots of possible mistakes to make, ii) there's not a huge amount of time to double-check them, and iii) I'm faster once I've warmed up a bit.
Also, for some of the stuff which I like to look at, it can be the case that once a proof seems to work, you can easily try to take "the hull of the proof", and let the result expand into the most general version of itself which will follow from the same arguments.
Following this path usually starts with a few explicit examples and a bit of pattern-matching, and it's harder to get those juices flowing when the clock is ticking. When there's room to breathe, you can iterate a bit more, get a sense for why each version doesn't quite fit, and so on.
The past few months were very academically social, with lots of good discussions and big-picture thinking, but not quite as much getting into the weeds of calculations, working through examples, and so on. It's been nice to hit the ground running on that front, and I'm hoping to keep that going.
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