No Country for Mediocre Mathematicians

Lying is a core part of communicating mathematics. We lie to kindergarteners when explaining fractions. We lie to fourth graders when approaching limits. We lie to the NSF whenever they question if our work is important. And, I lie to anyone who asks me why I’m a mathematician. It is much easier to claim “I love learning the laws of life,” while literally handwaving, than it is for me to flashback to the twenty or so pivotal moments that lead to me walking out of Gainesville with a PhD in Arithmetic Geometry. Unfortunately, career choice happens to be the prototypical conversation starter, so hundreds of times over the past year I have had to choose between an hour-long walkthrough of my childhood or a small fib, and I am yet to tell anyone of my days at puzzle camp. Lying, however, is a sin, and those small fibs add up to a big wrong. Hence, I make it a point to counteract my lies by always answering the inevitable follow-up question of “what do you plan to do with a math PhD?” honestly, with a shrug.

Other mathematicians answer easily with “math.” I’ve met career mathematicians before. They’re at conferences and universities and on twitter (sometimes). If you find the right corner of our sphere, you can stand inches away from the “smartest people alive” and revel in their brilliance. But strangely enough, every time I interact with any of these geniuses, the feeling I get is never an awe-filled “golly this person’s intellect dwarfs mine,” but instead an unblinking “wow this mfer really loves math.” Passion, talent, and ego intertwine inseparably in the world of academic mathematics, and my personal passion has always been scarce. When I first met with my advisor, Jeremy, he asked me whether or not I planned on pursuing academia, which I responded to in the negative. I figured it required momentum and ego beyond what I had accrued. Jeremy did not disagree, and so we set off to learn at a leisurely pace for four years. At the end of my doctorate, I had accidentally accomplished more than we both expected: solo published in a solid journal, presented at a big conference, and stowed some cash from a couple teaching awards too. Three decades ago this curriculum vitae would have been a golden ticket to almost any post-doc position of my liking, but it is impossible to live today three decades ago, and competition has done naught but accelerate. Jeremy and I were not mistaken to not believe in me.

We had factored the climate of the present day in, of course. We were aware that I needed triple the amount of conferences and maybe two more solid publications to guarantee a post-post-grad position. In the present moment, one where I’m flailing without a concrete job or future, I can’t stop myself from imagining where I could be if I had just produced a little bit more. Don’t be mistaken, I was not a terrible post-doc candidate, simply a weak one. I just graduated into a tough market for a mediocre grad student turned mediocre mathematician.

Jealousy is never a good look on a woman, which may be why my modeling career never took off. During the 2026 winter olympics, prodigy figure skater Alysa Liu won the gold medal in the women’s singles event and shot to stardom seemingly overnight. The twenty one year old’s passion for ice skating lit up the American public and dating rumors swirled as she was caught hanging around hyperpop idol glaive. And for some unbeknownst reason, I seethed with envy. Despite the fact that I have touched an ice rink twice in my life total, I lamented not having Alysa’s focused devotion to the sport as a youth. I don’t even listen to glaive and still I wanted to be her so bad. Jealousy sometimes strikes when you least expect it. Other times, it can be pretty easy to predict. Almost exactly a year before Alysa’s ascent, Hannah Cairo submitted her first preprint to ArXiv. Within a couple of months, the media and mathematical community took note and had become abuzz about her age (17), her clever counterexample to the Mizohata-Takeuchi conjecture, and her immediate application and acceptance into graduate school. I, a 26 year old graduate student still waiting on my first publication at the time, had no such commotion about. Comparison is the thief of joy, and I don’t lock my doors (I’m hyperstitioning a high trust society), so I get regular visits. Alysa, Hannah, and I did not share the same trajectories.

But from each according to her ability, from each according to her want. Mathematics is a vast frontier that all are free to explore and none are able to conquer. I will not accomplish what my advisor will not accomplish what Sarnak will not accomplish what Ramanujan accomplished. And though our contributions are not the same, they are, tautologically, contributions. My physicist friend once asked me what the point of doing research was if someone like Terence Tao could have figured out everything in my dissertation in a tenth of the time. I answered by pointing out that Terence Tao didn’t. Terence Tao did not find a small open problem posited by my advisor and publish a bite sized result making incremental progress. He has only so much time and so many other fish to fry. And in his absence, I was given the chance to touch the edge of knowing and experience the unmatched exhilaration of discovering structure in the labryinth of everything. In a tiny way, I really did learn of laws unseen by others, and in an even tinier way, that was important. Small ball mathematicians have always existed, and they’ve always been important. For every landmark theory, theorem, or conjecture, there have been incremental, partial results supporting intuition and inching towards the white whale. When I attended BARD, a small computational number theory conference, one of the organizers preached of the outsized impact we could have just by being willing to program the numerical experiments that other mathematicians only theorized about. The small ball player can completely change the approach and intuition of the leading names without ever joining their ranks. The mediocre mathematician has always had purpose.

Hence questions of my academic inferiority were easily dismissed. I would never be George Andrews or Andrew Wiles and I had no need to be. Cairo’s result could land on an appreciative and still heart. I could be excited for the future of the bright young minds that had found the brand new counterexamples for the Mizohata-Takeuchi, Jacobian, and unit distance conjecture. Right?

I hate talking about AI. It feels like I’m playing make believe with science fiction roleplayers, except I have to nod with complete sincerity else they’ll sense my disbelief and cast me out of the raid group. I hate hating AI too. An ignorant hatred of AI maketh me as a Trojan amidst Cassandra’s prophecies. And all too well, because I also hate prophecies. But often what we hate is what is to be done.

Mathematicians will use AI. They already do, and they will as well. Even if the reasoning models make zero improvement from today onwards, they will change the face of mathematics irreparably. Although, not every young mathematician knows this yet. March 2026, I joined some of my fellow graduate students at an Indian restaurant for an end of the school year/birthday joint celebration. When I arrived, they were taking turns scooping basmati rice from a platter balanced on the wedge of our bouquet of tables while deriding the silly undergraduate students who relied on ChatGPT for calculus help. Laughter doubly ensued when my underclassman described the time he asked the Google Search AI how he might enumerate the number of partitions with prime valued cranks and it hallucinated a gobbledygook approach summing the Catalan numbers. I quietly found an empty seat and stole some Aloo Gobi off my friend Emma’s plate (she later told me it was too spicy for her anyway). It’s painful how behind we were. AcerFur, an undergraduate at Cambridge, and Leeham, a self proclaimed non-mathematician, had been solving Erdos problems using GPT-5.2 since January. In May, OpenAI announced one of their models had found a counterexample to the Unit Distance Problem. In July, Levent Alpoge at Anthropic announced that Fable had disproved the Jacobian Conjecture. I could go on to describe tens of open problems that have been autonomously resolved and a hundred more genuine results that heavily relied on AI automation, but I won’t. It brings me no joy to describe the success of frontier models doing math more impressive than mine; it’s a thing I hate talking about. After I washed down the mild (sorry Emma) curry I stole with a gulp of water, my friend Josh asked if I had looked at the current mathematical capabilities of the silly AI. I lied and said that I hadn’t.

On August 9th, 2026, I pushed a preprint to the Math for AI Safety Repository, a preprint that I coauthored with Claude Opus. Before, automated proof generation had just been the muffled tears of battle in the distance, but that clamor quickly turned visceral when the front lines arrived at my doorstep. In fear, I must admit that, in our game of two, Opus may have been the most valuable contributor.

While I honed in on the niche topic of Artin-Schreier curves for the last three years, Opus secretly mastered (or at least read a textbook about) algebraic complexity. In this textbook, that Opus read, lay the crux of our argument, an argument that would have taken me weeks (months) to apply if I already knew which book to pick up. This, though, did not scare me. My co-mentee had already shared with me how ChatGPT’s literature search had found a niche lemma from 1971 that plugged a small hole in his dissertation. Breadth is the oft noted leg up that AI models have on humans. Overlooked is the speed at which it tests ideas. The journey of a proof is typically defined by the numerous rabbit holes that a prover naively wanders into, thinking it a mere fox hole. These human weeks of malaise and confusion (which tend to be remembered fondly in the retrospect) are condensed to days (hours) by the machine. Numerical experiments that would have taken me a whole day-night cycle to set up, run, and interpret fly by in the background with almost no need for my input. Easy incremental progress gets spit out by the LLM while I occupy myself with washing its Godawful prose out of a report in the foreground. Something I have to do because I begin to lose myself after reading too many claudisms and terrible unanchored AI intuitions. LLM output is legitimately exhausting to read. Which is a problem that Claude also solves while solving; it wanders tens of wrong avenues after I’ve already spent my limited supply of pure focus. While a girl can only do so much math in a day, Claude Opus is not a girl.

I don’t know if Jeremy felt the same way about garvyisms and my silly grad student guesses as I do about Claude’s, but thankfully he dealt with me anyway. He patiently watched as I slowly navigated wrong roads and backed down sketchy alleyways and would gently remind me to maybe try other directions. I am immensely grateful for his eye over my work for I would not have grown as much without it. Luckily for him, mentorship is not a one way street. Someone had to have the bad ideas and mark the map with red xes and I’m sure he was glad that someone was not him. Jeremy can only do so much math in a day too, and despite everything I remember those weeks of confused wandering fondly in retrospect. The dead ends I’ve marked are mine and mine alone.

When, in the midst of my first research problem, Jeremy asked me how much time I was spending with the beloved and requisite Artin-Schreier curves, I replied five hours a day. He raised an eyebrow and I revised the number down to a more truthful four. Under oath, I would have volunteered an even meagerer figure. I do not have the special constitution that allows one to stare at a single equation for a full work day (despite the accusations, I am neurotypical), not unlike the majority of mathematicians. This had never struck me as pure downside though. Pure math is a series of self-referential attention vortexes, each branching into an unending array of splintered paths. Fun problems exist and persist, independent of your location, and threaten to swirl you around tirelessly. It’s a beautiful coincidence, then, that we have a built in timer to tell us when life should happen instead of study. A timer that makes us human, and worse at math. A timer we can now turn off.

What constitutes as “mathematics” has long been debated. Is it a collection of related topics? Does software engineering count as a mathematical problem? Where does philosophy of math stop being philosophy and start being math? There is no widely agreed upon answer. Thus! I posit my own: Mathematics is the thing you try to understand, don’t, get frustrated about, and then do. It’s difficult to draw the connection in content between basic arithmetic and basic arithmetic geometry, but that process of internal discovery is a true commonality. The confusion a child feels when internalizing multiplication as repeated addition is the same that a graduate student feels when convincing herself that there really isn’t a quintic formula. This process is what connects all mathematicians together, from Ptolemy to garvy. We’re all frustration addicts. We just want to bang our heads against problems we don’t yet know how to solve. So of course mathematicians purport to care of the progress of human knowledge and the conquest of reality's rules. From where else would we get our fix of difficult problems? I’m sure there really are mathematicians who truly believe our cover story of righteous exploration and are genuinely excited about our new chance to accelerate our pace to universal enlightenment. It might even be most of us. After all, when you tell a story enough times it becomes something that’s neither real nor fake; it becomes dogma. So as a person of faith, when anyone asks me why we do mathematics, I am sure to recite the standard catechism without hesitation. I just sometimes forget to also cross my fingers.

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https://www.nbcolympics.com/news/alysa-liu-wins-olympic-gold-2026-milan-cortina-games

https://www.cosmopolitan.com/entertainment/celebs/a70573259/alysa-liu-boyfriend-glaive/

https://arxiv.org/abs/2502.06137

https://www.quantamagazine.org/at-17-hannah-cairo-solved-a-major-math-mystery-20250801/

https://openai.com/index/model-disproves-discrete-geometry-conjecture/

https://openai.com/index/ten-advances-in-mathematics/

https://math.mit.edu/~poonen/papers/M23.pdf (note that this is just one example, but it is really not too difficult to find more)

https://github.com/lionellevine/MAIS/blob/main/open-problems/O62/MAIS-O62-progress-1.md