Advice to a First Year Me

The first, and I think most important, thing that you need to know is that research is *hard*. Really, really hard. You've done an undergraduate degree already, and that probably wasn't hard. You probably sat in lectures following everything the lecturer said, and the longest you've ever been stuck on an exercise is probably measured in minutes, or at worst hours.

Research is not like that. You will spend months thinking about things and getting nowhere. You may think you've understood that, but read it again, and try to actually understand what it means, I do literally mean months, and I do literally mean nowhere.

Your progress will come in fits and starts, and the fits and starts will be separated by long periods of nothing. Months of nothing is hard work emotionally. Almost everyone I know who has done a PhD in maths has complained at some point about the long periods of nothing. This even includes those people who I actually have managed to give this advice to before they started, and it even includes myself, after having given out this advice. You will be depressed because you don't feel like you're achieving anything. Don't worry, this is normal!

The following is approximately a graph of my perceived progress so far in my PhD thesis. the time axis doesn't have a scale, but it is roughly linear, and spans 3 years. Note that the flat bits really are genuinely flat and, if I'm perfectly honest, the spikes probably aren't quite spiky enough.

Note that I'm *not* saying that the time spent doing nothing is literally wasted. Obviously you need to check which ways of proving something don't work before you can find the ones that do, and, perhaps more subtly, there are so many hidden connections buried in mathematics that any time at all spent reading any mathematics is quite likely to turn out to be useful to you one day (I'll have an example of this in another post I'm preparing for later in the week). But it sure doesn't feel like that when you're going through the periods of nothing.

You are now hitting your limits. You are going to be doing something that is genuinely hard work. This is going to feel like hard work. Be ready for it.