By Charles Kulick

Studying for qualifying exams is grueling. It’s much like the gravitational constant or the Feigenbaum delta or the speed of light; qualifying exams demand effort. No matter how much I loved math, and I certainly loved math quite a bit, the exam preparation process was enough to leave me exhausted. The summer before grad school, excited to launch a new chapter of my life, I thought I was ready.

I failed my first qualifying exam horribly. Out of eight questions, I only submitted one solution. Just like that, a summer of study had left me at a point still far below where I needed to be for a pass. I’d have to study again, and this time, there’d be multiple qualifying exams waiting to potentially fail me. What was I getting into?

Honestly, I didn’t know what to expect from graduate school. As a first-generation student from the small city of Scranton (yes, the same Scranton of The Office fame) I’d chosen to stay local for my undergraduate program. The support of family and friends helped smooth the transition, and the small department was incredibly accessible and welcoming. Starting graduate school on the west coast at a large university was a completely new experience. My abject failure certainly magnified the culture shock, and it left me questioning if that passion I’d felt for math was ever really… real. The road from losing that drive to total burnout is short and strictly downhill. That’s my goal here, today: sharing the tools I’ve learned, both from myself and my peers, to regain that central meaning and purpose in what might be a completely new academic situation.

Why was it math in the first place?

Everyone loves a good origin story. The reason we begin to pursue mathematics is often a simple interest in a particular topic. When I was an undergraduate, cryptography and coding theory were those topics that pushed an interest into something more. Cryptography in particular was the first place I saw math as something more than a necessity; it was a branch that stood on its own, unique in its application, yet derived its strength from algebra and number theory and more arcane studies besides. It was magical! While our interests evolve, as did mine, returning to explore the things that once excited you can turn out to be even more fulfilling the second time around. A great tool to facilitate this can be a Directed Reading Program (DRP)! As Christopher recently wrote, it is a great opportunity to allow you to explore a topic with an immense level of detail and care. If your program has a DRP established, it can be as easy as signing up to be a mentor. I signed up and advertised an interest in a cryptography reading group, and sure enough, my two mentees Anna and James found me. From cryptography, coding theory, and quantum computing, to understanding the undecidability of the Continuum Hypothesis and how to solve a Rubix cube, interacting in a small and personal setting allowed all of us to explore math in spontaneous ways not easily facilitated otherwise. Both the student and the teacher benefit from an experience this uniquely wonderful. In teaching to others, I had returned to a core area of interest and witnessed it imbued with new life by the valuable perspectives of my students. This was an incredible way to rediscover my motivation; these situations are uniquely enabled by the joy of shared community we find in mathematics.

When are you most excited about your work?

As an undergraduate, I never would have imagined I’d be anything other than a pure mathematician. At times my interests were more algebraic, at times more analytic, but never really applied. I was more than content to simply enjoy mathematics. Reexamining my journey, however, opened new perspectives. Was it really the algebra of coding theory I enjoyed, or the immediate connection to applications? Was cryptography fascinating in the abstract, or more in the concrete? With a change of perspective, I finally realized it was the latter. Near the end of my first year of graduate school, I completely changed my path. The switch was amazingly smooth; with the advice of knowledgeable mentors and kind peers, I was quickly integrated into the community. One of the greatest benefits we have as graduate mathematicians is a remarkable connection to so many other people studying every esoteric mathematical subject known to humanity, whether in our departments or even just in our wider friend group. If something is interesting, asking about it is a very low risk and high reward action! You might find yourself excited by the things you hadn’t yet bothered to try. Going to seminars, talking over the research of friends, and frequenting conferences are all great ways to keep this exploration going. While our favorite topics can excite us, missing out on the perfect problem can only happen if we’re willing to let it pass us by. It’s so easy to believe the gulf between disciplines is too large to cross, or that we’re already too specialized to change focus. But keeping an open mind can be crucial to finding the right place – sometimes, that local maximum isn’t global, and you have to be willing to explore just a bit more to find it.

In my case, once I realized that connecting my work to the world was what really gave me meaning, I found myself interested in machine learning. In particular, I started trying to understand the theoretical underpinnings of strong systems and thinking of ways to approach safety issues like agentic alignment. This was a new level of fulfillment; my study felt connected directly to things I cared about. This also led to some of the most interesting conversations of my life, as I found people who enjoyed the same kind of work. Noticing the causes of your excitement can be hard, but of all ways to regain a passion, pinpointing this might be the most crucial.

If you could only solve one problem with your life, what would it be?

The natural follow-up to this question, whatever your answer, is: why not work on that problem right now? I’ve been guilty of a great deal of mathematical procrastination. Even after realizing my interests were applied, I felt underqualified. As a first-year graduate, I didn’t even try to do research in the field; I could barely study for quals while trying to pass my classes, so surely I wasn’t ready. It wasn’t until recently that the lesson of the abysmally flunked qual truly clicked – you really can’t improve until you fail. I might not yet know all of the math I’ll eventually need, but really, the same is true for every area of research. Diving in can lead to incredible places, and will inevitably lead to improvement much faster than watching from the outside. If you find yourself procrastinating on a goal in your personal life, the common wisdom is to pursue it immediately or else risk giving it up forever. This should not be any different in mathematics. When what you want to work on and what you’re currently working on are in conflict, it’s very natural to feel strained. By reconciling your desire and your trajectory you can regain the intrinsic motivation so necessary for long-term success.

In the end, I found my passion again. I still have many years to study, but it’s clearer now than ever how very important it is to keep that meaning alive. And with the wonderful community of mathematicians I’m lucky to call my friends and peers, I’m confident it will be flourishing for years to come.

Charles Kulick is a Ph.D. student at the University of California, Santa Barbara studying machine learning. He lives with three cats, and bowls in his spare time.