I haven't really written about that! Those are deep question. At the start you are a PhD student so your supervisor gives you problems. Then every time you achieve something it opens up new problems. One of the skills is to recognise what might be interesting and fruitful. 1/n
2/n Personally I periodically write lists of things that have occurred to me to think about. Often I'll do an initial investigation to test the waters, and then decide what to think about first. Often it chooses itself for me because I can't stop thinking about it.
When it comes to choosing a strategy: that's often how I pick which question to think about next, because I actually have a strategy in mind. If I have no strategy in mind, just an interesting question, that is quite likely to come to a dead end quickly. 3/n
Strategies often come from noticing similarities with something else you've seen, and either generalising it or modifying it for the different setting. So spotting similarities, especially non-obvious ones, is an important skill in my work. I do it the whole time in life too. 4/n
I'm not sure where I got that skill, or how one can learn it, but it's how I teach too - by using a lot of analogies. Sometimes building a strategy involves reading around to see what sorts of things other people have done in similar situations. 5/n
I think it's more or less impossible to estimate how long something will take, at least in my work. The problem my supervisor gave me for my first week of PhD ended up being practically my entire thesis! I recently proved something that took me 10 years on and off. 6/n
I thought it was going to be pretty routine, like maybe a month! The only time we really ever have to estimate how long something is going to take is for grant applications when they ask us to draw diagrams of timelines of when we will do what. You just have to make stuff up. 7/n
It's one of the things that is probably geared more towards experimental scientists where experiments might have fixed lengths. Of course, we have to estimate how long something is going to take if we're under pressure to publish papers regularly. 8/n
But in pure maths I'd consider 1 paper per year to be fine, and more than that I think is a lot. As for how we measure progress: in some fields it's really just proving big theorems. But in my field there could be more to it. Sometimes it's about making a good definition. 9/n
Then the aim is to demonstrate what is good about the definition, the aim isn't exactly to prove things. Although proving something might help show that it's a good definition. But it might also be showing that it unifies a wide range of examples. 10/n
If I think about what I reckon are the greatest contributions to category theory, quite a lot is in coming up with really fantastic definitions giving us ways to think about structures in a new way. This is the sense in which I say maths is not just about right and wrong. 11/n
It's really about making connections between things and giving us new, productive, illuminating ways to think. 12/n
From a more practical point of view we judge progress by talking about our work with other researchers and seeing what they think. We give talks at seminars and conferences to test it out and gauge people's reaction, as well as for general dissemination. 13/n
We also think about whether something is enough to make for a journal article. Usually you either need to prove something new about existing structures and show that the result is helpful to something else. Or... 14/n
...or propose a new definition of a structure, show that it unifies some interesting examples and has some interesting properties. 15/n
In my most recent conference talk my work was to give a more abstract way of thinking about something very well known, and show it thus encompasses some more examples including helping me prove something that has been puzzling people for a while. 16/n
I was gratified that even people who weren't necessarily interested in the thing I was trying to prove were interested in the new way of thinking about the well-known thing. So now I'm writing up the paper. I think this is all I have to say for now. Thanks for asking! n/n, n=17.
