One of my mathy friends recently said, “I thought that when I finished graduate school I would have learned pretty much all the math I’d ever know.” When I was getting ready to graduate, I felt the same way. Which was a bit scary, because I felt like I didn’t know very much. But I figured that my future research life would be applying the techniques I used in my thesis to similar problems. I imagined that I would pick up some new ideas and go deeper into the area, but that my baseline math knowledge was pretty much set. This has been entirely wrong.

In fact, I have found myself working more on problems that I have absolutely no idea how to solve, and have realized that my interests refuse to stay bounded by the math that I know best. And problems don’t care! They may seem like problems in number theory, but oh, suddenly here comes representation theory, and a whole bunch of group theory! So I find myself trying to learn whole new areas, or relearn things I saw in courses but couldn’t really absorb at the time. It can be really hard, with work and so many other things going on.

I have this sense that if I had been a better graduate student, everything would be so much easier now, and that I should probably just take a few years off and go back to school. It would be great to be in a structured environment, designed by experts to help me learn more math. In my post-thesis research life, I find that so many topics from my classes really did turn out to be important, and those that seemed irrelevant at the time (I mean, when was I going to need measure theory?) have turned up over and over again. And I constantly wish I knew stuff from classes that I couldn’t bring myself to take (classical groups, linear programming… the wish-I-knew-list grows every day). But I can’t go back to school. I have a job now! I wasted my chance! Nooooo!

Of course, I have to remind myself that I actually learned a ton in graduate school. And at the time I was stressed out, worried about the very immediate problems presented by my thesis and teaching, as well as eventually finding a job. I was the best graduate student I could be, given who I was and the time I had. So what do I do with this wistful sense of what I should have been? I just keep doing the best I can, trying to solve problems, learning things from books, going to talks, and asking people questions when I get the chance. And, slowly, I do keep learning new things and solving new problems

The regretful sense that I should have been a better graduate student is increasingly countered by a feeling of excitement in the realization that I am actually still growing as a mathematician. In fact, as I (through great struggle) learn more, it becomes a little easier to see bigger pictures, and I even learn faster. Graduate school was just a jumping off point, and I keep gaining new perspectives. I am so glad that I was wrong about my math trajectory! It is actually thrilling to realize that, far from being done learning, I can just keep getting better at this.

I don’t know why I didn’t realize it sooner. I mean, I guess I thought my professors just started out good at math; that they were of a different species of mathematician. I definitely didn’t think of them as still learning. Perhaps I had, without even knowing it, absorbed the idea of mathematics as a sort of inborn talent, that you could fulfill as a young person but which fades as time goes on. Thank G. H. Hardy, with his incredibly annoying statement in *A Mathematician’s Apology*, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.” Perhaps Hardy’s words applied to his life. After all, he was a Tripos star, at the top of his field from the very beginning. He may have just crammed in a huge amount of math early on, and never really felt that he was growing in the same way later in life. However, I definitely followed a different path, and many others have as well. Richard Guy, anyone? (Happy 100^{th} birthday!) In fact, I am not the first to point out that this whole idea is suspect. I reject the notion that these early years are the peak of my career, and that people’s mathematical value or potential is determinable by their 40^{th} birthday. Hooray for early achievers! But also, hooray for people who start later and move slower but keep getting better all the time.

Realizing that I can keep getting better as a mathematician has made me really empathize with my students, who often think that they are already basically good or bad at math, and are afraid that they have hit their limit and are not capable of understanding some difficult concepts. This takes me back to growth mindset, (which I just explained to my linear algebra class on the first day of class!) and how easy it is even for me to fall into the trap of thinking that I can’t grow in some way.

On the level of the profession, it is important to reach out to talented young people from all backgrounds, both to maximize mathematical progress and to make the mathematical world a more diverse place. However, I think that it is also important to keep the door open for those who are not early talents, who may not choose mathematics until much later but still have a lot to offer. This is vital to broadening the profession, since despite our best efforts, students from underrepresented groups in STEM or less affluent school districts may not come into contact with interesting mathematics until later in life, and may have far fewer opportunities to pursue mathematics young even if they are interested.

So that’s what I’m thinking about as I start the new semester. Good luck to everyone with the challenges of fall! I would love to hear your thoughts on all this in the comments.

Thank you for this excellent post! I think that your point is so true that the attainment of your PhD and the knowledge you have gained initially seems like a pinnacle but there is so much more learning and growth along the continuing trajectory. This is certainly true in math and academia, and also, I think, in so many areas of life. May you someday know a blue whale of math!

“I have this sense that if I had been a better graduate student, everything would be so much easier now, and that I should probably just take a few years off and go back to school.” I’m glad that everything else you write here contradicts this wrongheaded idea!

Many of the things I teach my PhD students as absolutely basic, foundational calculational tools are things I learned post-PhD. (Some of them things that my advisor thought I should learn, but wasn’t that insistent about at the time…)