Let me start out by apologizing for two things, first the horrible pun in the title, and second my absence from the blog for the summer. Between moving twice (once cross-country), graduating, getting set up at a new job, buying furniture, trying to finish some papers, and being academic coordinator at Mathcamp I was pretty swamped. As a result I missed out on some developments in the math blogging.

Frequent commenter Danny Calegari started a blog in May. It pays to occasionally click on the links in comments here as sometimes you’ll find brand new blogs. My mathcamp friend, Matt Kahle, who is a postdoc at Stanford also started a blog. It has a fun mix of some elementary stuff (like the Rubik’s cube) and some of his research (which as an interesting mix of topology, combinatorics, and statistical mechanics, it definitely involves a lot of sending n to infinity in ways that would make my advisor happy). I’ve been meaning to link to both of those since sometime in June but just haven’t gotten around to it (though I did manage to add them both to the blogroll). It’s been that sort of summer, just ask me about my passport. Also, low dimensional topology has become a group blog. I find group blogging a great model both as a reader and blogger because it promotes conversations and allows one to maintain a reasonably updated blog even when someone disappears a whole summer.

Finally, over the summer there was a great conversation about what mathematicians need to know about blogging. Here’s my two cents. One thing incredibly valuable thing about blogging is the opportunity to have discussions and get advice about how to be a mathematician. It’s often hard in real life to have a discussion involving people at many different places in their careers about professional questions. In that spirit, here’s a question I’ve been wrestling with lately. How do you balance your research time between the following three activities: working on problems you basically know how to solve, working on problems you don’t know how to solve but are important problems, and learning new tools. When I was in graduate school I felt like it was pretty easy to balance things because any time I had any idea that was at all worthwhile I just worked on it and when I didn’t, I learned new things. I had few enough research-worthy ideas that it was feasible to think about all of them. Now that I know more I can’t keep doing that because I simply don’t have time to work on all the easy problems that I could solve. So the need comes to prioritize. I was wondering how other people strike this balance.

There’s another category in addition to the three above: writing up problems that you have already solved. I’m sure you’ll get more in that category as time goes on.

Somehow paper writing goes into a different category of activity for me. It takes a different sort of energy and a different sort of concentration to write than to work on a problem, and so I write as often as I can force myself to, and that’s the balance I strike there.

When you have more problems that you would like to work on (and have the ability to make progress on) than you have time for, then you have the option to collaborate, pass on your problems to colleagues, or take on a graduate student. or you can just add it to some private list of problems that you can turn to when you feel unmotivated to do anything else. You can also try to sneak in an open problem or two into each paper to encourage someone else to take it up.

I tend to spend a fair bit of time on “mini-problems” which are not necessarily publication-quality (e.g. can one prove standard lemma X using method Y rather than standard method Z?), but not quite within one’s ability to solve either, and are possible precursors to fully fledged research problems; these tend to be a nice combination of the three activities you mentioned. (I talk about this a bit here and here.)

I try to write up things as I solve them so they’re mostly written up by the time I’ve figured them out. This doesn’t work with expository stuff, and it doesn’t always work very well with collaborative papers… but when I can do it, I’m happier. I don’t like writing up stale old stuff.

[i]I try to write up things as I solve them so they’re mostly written up by the time I’ve figured them out.[/i]

I try to write up things as I solve them too, but I don’t succeed.

When I was starting out I explicitly tried to spend half my time working on big problems I didn’t know how to solve, and half working on problems — well, maybe not that I really _knew_ how to solve, but that at least I had a reasonable strategy for. More recently I’ve found that this happens without my needing to think about it explicitly. As for learning new tools, I do it almost entirely when a problem demands or at least suggests it, and this (at least for me) seems to be the right amount. (Relevant anecdote: at some point in graduate school I told Joe Harris I wanted to take a year off so I could stop working on my thesis and just read EGA and really learn algebraic geometry. He told me this was the stupidest idea he’d ever heard. For me, this was good advice.)

If I were to change something from my previous system, it would be to follow John and write things up more carefully as I work them out, as I do now. Now my partially finished projects are .tex files that look like papers with gaps, not unreadable sequences of handwritten pages. It’s a big improvement, especially when you have a bunch of graduate students with whom you want to share ideas.

(tenured, 10 years post Ph.D.)

So this explains the spike in my traffic.

I should remark that “my” blog is also a group blog, albeit with a very shy (up till now) co-blogger. I will resist the temptation to out him/her in this forum, but encourage them to out themselves (perhaps as a spur to activity?)