A piece of advice for young graduate students (in mathematics) that I've heard frequently is to find early a "side project" to work on alongside your main research enterprise.

An obvious question that follows is: what are the characteristics of a good side project?


closed as off-topic by user223391, Matthew Conroy, Daniel W. Farlow, Did, Stefan Mesken Dec 24 '16 at 22:25

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    $\begingroup$ Yes, but the things which make a good project good may be invariant between topics. If you think that's not the case then you should have some specifics to back up your point. You could include these specifics in the question. You could also give examples of side projects which you've heard about and point out how they satisfy these specifics. You should also probably point out how topic-sensitivity is not too prohibitive. That is, you claim that biology and mathematics deserve different questions. What about complex analysis and set theory? Do they deserve different questions? Why not? $\endgroup$ – Will R Dec 24 '16 at 20:08

I'm certainly an advocate of this approach. Here are the criteria I consider important for a side project:

  • It should be fun.

  • That's it.

Here's the unfortunate fact: as beautiful as mathematics is, and as much joy as it can bring you, by the very fact that you care about it so much it will cause you truly epic amounts of stress. And the fact that as a grad student you're surrounded by lots of brilliant people with far more experience than you (heck, your advisor should be one of these people!) doesn't help.

In my mind there are two things a side project can do: it can broaden your horizons mathematically, and it can be something that keeps math fun. Although the former is valuable, it pales in comparison to the latter (at least, in my opinion). Nobody really knows a surefire formula for getting mathematical inspiration, but being happy never hurt. Make your side project something that can remind you why you love math, even if your more serious work is going poorly.

And, more predictably than helping you get inspiration, a fun side project helps you do math every day. Math is like any other skill: practice matters. There are definitely days when one is too stressed out to work on their main project; well, it still works the brain-muscles to fool around with some funny piece of math over yonder.

Incidentally, lest you doubt my seriousness above, my favorite side project came about from a conversation with my advisor which went essentially like this:

"Can I do this with ultrafilters?"

" . . . don't do that with ultrafilters."

"I'mma do that with ultrafilters because reasons."

And it kept me focused and happy throughout my first "serious" project. I am a firm advocate of the value of silliness in mathematics, especially in grad school.

  • $\begingroup$ Smiled two times while reading. Can't give it a +2, unfortunatelly. $\endgroup$ – lisyarus Dec 24 '16 at 19:34
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    $\begingroup$ When I was younger, `I am going to' was pronounced Onganna. Times change $\endgroup$ – Will Jagy Dec 24 '16 at 19:59
  • $\begingroup$ I like this answer very much (+1). However, to add to the point I've made in other comments, I don't see how this is really mathematics specific. What's stopping me from changing every instance of the word "mathematics" in this answer to the word "biology" instead? [of course, the fact that this answer applies so generally is probably a good thing, but my point is about the question's place in math.SE] $\endgroup$ – Will R Dec 24 '16 at 20:10
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    $\begingroup$ @WillR That's fair. I think the main difference which sets math aside is the ease with which one can invest in a side project: no experiments or fieldwork are required, no documents need be read (of course, the more one reads the more one will get out of it, but you can have a side project conducted completely independently of the rest of the world). Mathematicians can be "flippant" about their side projects in a way most disciplines (to my understanding) can't. To my mind, this makes it more okay to do a side project purely for the fun of it. $\endgroup$ – Noah Schweber Dec 24 '16 at 20:16
  • $\begingroup$ +1 I appreciate that you bothered to put an apostrophe in I'mma. $\endgroup$ – Servaes Dec 24 '16 at 21:17