Here’s how it happens: You’re in graduate school and were one of the best people in your major from your school. Honestly, that’s how you got into graduate school in the first place. You go in the first few weeks, you meet your new peers, and you engage in mathematical discussion. It’s really fun, being with people who are just as excited about math as you are. But then, a horrible thing happens. Someone, in conversation, mentions something you don’t know. And not only that, but the way they talk about it suggests that anyone who knows anything about anything knows what they’re talking about. Or maybe, in an even worse turn of events, this person is a professor. What are you going to do?
This kind of phenomenon seems to happen to people in every stage of their mathematical career after meeting new people, but it appears to be especially prevalent among graduate students. I think it’s due (at least in part) to the fact that the math that students learn in high school and early college tends to be somewhat standardized, but there’s a lot you can learn in your later undergraduate years and early graduate years that other people just don’t come across.
I’ve pieced together advice I’ve gotten from various people and compiled it here. Being a math person, I have decided to divide it into two cases (although half of the problem with imposter syndrome is actually not knowing which case you fall into).
Case 1: You are not the person that knows the least in the room.
In this case (which, according to anyone I’ve spoken to—professors, other graduate students, etc.—is extremely likely), getting over imposter syndrome simply comes down to recognizing the assertion that it is tremendously unlikely that you are the smartest person in the whole world so eventually you are going to run into someone smarter than you. Moreover, graduate school is the bridge into real mathematics where knowledge is much less “well ordered” like it is in undergraduate mathematics and much more “partially ordered.” For example, I had a professor for real analysis in my undergraduate career who had forgotten all of his algebra knowledge. I mean, all of it. We knew this because some number theory actually came up in an analysis problem (believe it or not), and he had marked it as a “Challenge” problem in our homework when really it was the first lemma in our algebra textbooks. Now, am I smarter than this person? No way! I won’t mention the person by name, but there is exactly a 0% chance that I am anywhere near his level of intelligence. The point is that honestly, there’s a mistaken belief people seem to have that the concept of “intelligence” is well ordered.
Case 2: You really are the person that knows the least in the room.
Okay, so you know the least amount of mathematics in the room. Woe is you. Who cares? Sometimes it can actually be an advantage to be the person that knows the least in the room. That way, everyone has something to teach you. Flipping that around, you can learn something from everybody!
Additionally, you don’t need to be the cleverest to do mathematical research. I (and many of my peers) fall victim to this idea in research of “Why am I even trying this when [name of prominent famous mathematician] could have this done in a week?” Well, the real answer to that question is, your prominent famous mathematician doesn’t have the time or the interest to work this out. There are other things that person is tackling. Further, just because you’re the (hypothetical) least smart person in the room, doesn’t mean you can’t or shouldn’t solve a problem. It is well documented that math is about persistence, not “genius.” (See this post by Terence Tao for an interesting expansion on this subject). Plus, once you solve this problem, you become the sole person in the world who best understands your problem and maybe that will yield problems in the future that you are the best person to solve!
Everyone needs to cross the bridge at some point in their life where they learn that they’re not the best. Hopefully these tips lead you to recognize and start to get over your imposter syndrome just a little more easily. (But really, I don’t know what I’m talking about. Sometimes I feel like I’m the least qualified person to talk about imposter syndrome.)