Old post (written as I finished my dissertation): With apologies to mathematicians who actually juggle, I am wondering about how people maintain multiple mathematical projects. There are a lot of questions I'm interested in. While some stay in my narrow field of mathematics and are continuations of dissertation questions, others veer into related areas, with contributions from as far afield as combinatorics and PDE.
How many projects are you comfortable with at once? Given the time needed for teaching, service, and administrivia, how many problems do you like to have in active rotation? I am reaching for three. I find that I get very wrapped up in one problem at a time and have an extremely hard time pulling my head out of it to look at another, even if they are closely related. My mind always wants to go back to the one it's spent the most time on recently. I have a little file of questions and I've got six at the top, which means I am keeping my eyes open for papers that apply and people who might be able to help out. I read somewhere that Conway liked six. Maybe over time I will be able to switch more easily from one problem to another and make steady progress on several...
Do you like to have different problems in various stages? Some folks seem to like having a problem they're just starting to think about, a problem that they're making active progress on, one they're writing up. Do you aim to have a range like this? Does it work? How do you do it? I guess right now, by default, I have some things I'm wrapping up (dissertation) and some projects I'm starting.
On a related note, I am getting some advice from my advisor (if I understand correctly) that I should not be spending much time starting different things right now --- I ought to concentrate solely on finishing. If anyone has any first-hand advice they'd like to share, I'd be eager to hear it!
New commentary (after time as something approximating a real professor): These days I am fairly comfortable having three or four projects in the air at once. I only manage this because they are closely related. I really have more projects, but I am not making any progress at all on a few of them -- it seems like the ones left behind are mathematically different than the others. I also manage the projects that are progressing through collaboration. That makes it a social affair and something that must be scheduled into my daily calendar. When a student emails to ask about time to meet outside of office hours it is easy to say, "Oh, I guess I could reschedule my research time and talk to you about calculus instead"........ unless you've got a Skype date with someone across the country!
Small weekly goals and discussions help me move forward with multiple projects even if I can't devote hours a day to anything. Just keeping my head in a problem on a daily basis is pretty important. Being an overworked participant in the American liberal arts system has pushed me to learn this.
On that note.... I need to escape my office as I have a whole free hour. Time to do some research!