The scene was a post-seminar dinner at a homely Vietnamese joint. We had just listened to Jacquet speak at the University of Chicago. A soft-spoken grandfather-like figure, he enjoyed the nearly universal privilege among the elite mathematicademia of mononymity, being referred to by his last name, a name which is immortally attached to dear number theoretic constructs like the Jacquet functor. My friend, a graduate student, asked me to pass the spiced rice, as we discussed the brilliance of a friend of our advisor, an Indian mathematician by the name of Akshay Venkatesh.

Our advisor recalls, “I remember talking to Akshay about a certain trick I had found in evaluating a particularly gruesome p-adic integral. I had not finished erasing the board, preparing him with a rough explanation before I seeded the details with chalk, when he interrupted me–‘oh I see! Then you just used Cauchy-Schwartz didn’t you?’ ” This last quotation was delivered with fervent admiration, a tone of voice reserved solely for the few individuals who had impressed him so deeply that their quickness seemed less like the product of brain substrate computing away, and more like raw, inexplicable, beautiful magic.

Yet there is no magic. I empathized, “The stories you’ve been telling do make him look brilliant. Yet the brain is nothing more than a very advanced pattern recognition and association engine. I acknowledge certain people should rightly give us grand impressions, but we do a disservice to ourselves by stopping there. The interesting question, to me, is to consider the chain of experiences, teachers, read books, contemplated problems, and muses and introspections that Akshay had to pass before being able to see things so quickly and clearly.”

A very delicious discussion followed, almost as appetizing as the bún thịt nướng I had ordered. Arthur C. Clarke once remarked, “any technology sufficiently advanced is indistinguisable from magic.” I contend any chap sufficiently well-educated and self-ruminating is indistinguishable from genius. It had become clear there was a dichotomy amongst us, some preferring this earthly explanation, while others like my friend Sarah maintaining “there are some people who can think orders of magnitude quicker and better than others can.”

Srinivasan Ramanujan is a prime example from history’s drawers of inexplicable prodigy. The story goes he taught himself mathematics from an engineering handbook, a compendium of five thousand formulas and theorems which were stated as facts and without proof. It is said Ramanujan worked out all or most of the proofs by himself, beginning with the easy ones and progressively working up to the harder until the process became subconscious. He could see the results that were true without ever fully being able to explain his convinction. Still today, we are unraveling the majesties found in his notebooks [1].

In the past, our only impressions of these intellectual paragons were of their cultured state, after they had progressed through the tedious process of developing their trade. In the internet age, we can do much better. Consider if Terry Tao had blogged about his learning process. Although it is difficult to predict from an early age if someone will be a great mathematician, I will attempt to make my point by considering what I believe to be the three safest bets, Qiaochu Yuan, Akhil Mathew, and Eric Wofsey. All three are around twenty years old, have shown great mathematical promise, and most relevantly, have some recorded history of their learning process.

Three crucial aspects shaped their success. The first is relentless curiosity, coupled with the ability to fulfill it. A cursory reading of Qiaochu’s posts on Math Overflow and blog [2] show that he takes an interest in nearly every aspect of mathematics, teaching himself at least the foundations before turning to other problems. For example, graph theory is not considered “boring” or “too applied,” as those who focus on high abstraction mind find it, and conversely, nothing ranging from infinity categories to elaborate spectral sequences is considered too abstract. Currently, he’s teaching himself quantum mechanics from a mathematical flavor. This shows a burning desire for breadth is foundational. However, the curiosity should be deeper than just the desire to learn as much as possible. Reading through his Math Overflow posts in chronological order shows a strong will to re-integrate and re-think the ideas he’s learned in the past in more general frameworks or how they relate to other areas; a consideration of edge cases and variants; and an overall attitude that can be summed up as: if there is at least one mathematician that has found some thing interesting or profound, I will take his emotions for that thing and make them my own.

In a similar vein, Akhil’s blog is almost primarily his exposition of various mathematical fields intended for his personal benefit–a young mathematician’s diary made public. Eric Wofsey posts FLMPotDs–fun little math problems of the day–nearly weekly (and in the past, daily), and a read through his blog’s history shows his musings about spectral sequences and arcane topologies. So far, this is fairly intuitive: to be good at something you have to practice it every which and what way, and in the case of something like math, question, rethink, make connections, drop hypotheses to understand why theorems hold, consider edge cases, and immerse yourself so deeply your dreams become math problems.

More importantly, however, this curiosity needs the ability to be sated, or it will wither. In the past, this was difficult and there are correspondingly much fewer mathematicians per capita historically than in modern times. But new resources make this problem obsolete. [3] [4]

The second important factor is social. Isaac Newton famously reminds us that no one is self-made. Qioachu Yuan and Akhil Mathew had incredible resources in high school, and Eric Wofsey had Math Camp. A surplus of mentors and encouraging peers seems critical, but because it is so hard to control, we will not look here too carefully. This applies equally well to the last requirement, namely a base level of analytic intelligence. Multiple studies have shown that beyond a certain point (around two to three standard deviations), further IQ is not particularly helpful. Instead, successful academics exhibit exceptional social skills. In the case of mathematics, if you were able to fluidly learn epsilon-delta analysis in the sense of baby Rudin, you likely have all you need.

The above three factors explain equally well Andre Weil’s procurement of an immortal PhD thesis at only twenty one years of age and Drinfeld’s phenomenal proof of Langlands correspondence in the n = 2 case for function fields. There is nothing more, no magic. Only the first factor is directly within our control, so we finally come to the part applicable to you.

Gian-Carlo Rota wrote in his memorandum Indiscrete Thoughts that Sitzfleisch is often a much better predictor of an aspiring mathematician’s future success than any other factor. The German word for “the ability to sit in one’s chair doing gruesome work for hours,” it is a perfect skill to be coupled with fervent curiosity. In the comments on Qiaochu’s About page, he confesses that the sole interest he actually puts time and effort into is mathematics, and not much more; unfortunately, this laser-sharp focus is a requirement, at least in an early stage. Akrasia is rampant in academic communities, and much has been said on the subject. My personal recommendation is that if you have trouble getting things done, I suggest skipping David Allen and using what has been to me much more helpful in my mathematical development than any single graduate text, the Power of Full Engagement. (Incidentally, it is very difficult for a person to develop with both a high level of base intelligence and a sickening work ethic. The former makes them undervalue the latter. Indeed, it took me three years of graduate school to painfully realize how little of my potential I unleashed without the Sitzfleisch ingredient. This combination alone over time gets you 90% of the way to world-class success.)

Mathematics is a subject older than most, and an appropriate level of sophistication has developed. Keep in mind that learning the tools of abstract algebra and measure theory is much more difficult than picking up etale cohomology or rigid analytic geometry given you have a solid foundation in scheme theory. In his TED Talk, Salman Khan explains how analytics tools for measuring children’s education showed most kids later to be deemed precocious spent an unusually long amount of time on certain basic concepts, then sped through the rest. As my advisor pithily puts it, “mathematics is not about understanding complex abstract ideas, but about understanding simple things well.” The rest is a corollary. For example, when I was 11, I spent my summer writing out five full notebooks of exercises from my precalculus book. As a result, the majority of undergraduate math, physics, and computer science seemed easy and natural to me, and I did not have to devote as much concentrated work on a subject until I tackled real analysis. (Two other critical leaps you must make are category theory and scheme theory. Afterwards everything will seem much easier.) For those not familiar with higher mathematics, consider that learning Japanese is much harder than reading a dense Japanese political text given you know the language.

“Genius” was an idea that the layman would marvel at, an incomprehensible leap in human mind that seemed more surreal than relatable. Like much of science, this too has now been daintily plucked from the “magical mystery” basket and gently placed in the “oh that makes sense” zone. We now understand that the third ingredient, raw intelligence, is necessary only as a dough, but it is the first which decides how many theorems the recipe makes, and how delicious they taste.

References

[1] http://www.imsc.res.in/~rao/ramanujan/contentindex.html

[2] http://qchu.wordpress.com/

[3] http://mathoverflow.net

[4] http://library.nu/