Jim Simons — of "Chern-Simons Theory" as well as "being a billionaire because he started Renaissance Technologies" fame — is, despite being both a successful mathematician and a (very) successful hedge fund manager, surprisingly humble. At the very least, he's surprisingly self-aware.
In particular, he has said on multiple occasions that as he rose through the academic ranks, he knew he was not the best mathematician. He knew he was very good at math, but his exposure to the very, very best in the world made him quite confident that building a life dependent on being one of the top 0.000001% of mathematicians in the world was not going to be his path to success. Keep in mind, a lot of what he's talking about at this point in his life and career is about academic success — that is to say, these realizations were dictating how he approached his career as a mathematician, not (later) as a hedge fund manager.
What he did realize he had an uncanny talent for, however, was having a good taste in math. Now, "taste in math" is a difficult thing to quantify, but it's definitely a real thing. Some people just know how to ask the right questions — questions that have a high "surprise to effort" ratio, and/or questions that impact a lot of peoples' work.
Note that neither of those two criteria (surprise-to-effort and volume-of-people-affected) have much to do with "difficulty". The hardest questions are reserved for the very, very, very best, and they get tenured positions for answering those questions and proving that those answers are correct.
However, you don't have to be the very, very, very best mathematician in order to ask (and hopefully answer) some really insightful questions. You have to be really good, but mixing in a little bit of "taste" can propel ones career just as much as being "the very, very best" can.
But this is not an article about mathematical taste — though that would be a cool article to write. This is a more general article, motivated by that specific example. Namely, this is an article about how we (as mathematicians, and as humans) can and should always be looking for ways to "play to our strengths". In Simons' case, he mixed "being in the 99.999th percentile of mathematicians" with "good mathematical taste" and used those to get to similar heights as those who are in the 99.99999th percentile (note the addition of a couple 9s there).
We can learn a thing or two from this.
And Also, Sports
Another place that we see this type of mixing-and-matching of strengths is in sports. (I hope that by now you're not surprised to see sports analogies in these posts.)
Consider the set of all NBA players. This is your 99.9999th percentile of basketball players worldwide — this is where Jim Simons lives. Now consider the set of all All-Stars. This is roughly 10% of all NBA-players, so this is the 99.99999th percentile of basketball players. Now consider the set of all Hall-Of-Famers (HOFers)— another "order of magnitude" hit. So now we're in the 99.999999th percentile. Finally, let's talk about those who are in the GOAT discussion — this is the 99.9999999th percentile.
But, does the NBA really just consist of the 99.9999th percentile of the basketball world? I.e., if we could order every basketball player in the world be some set of tangible metrics, does the NBA just consist of the numbers 1 through 400 of that line-up? No.
The converse is true: if you're in the 99.999999th percentile, then you're in the NBA. But away from this teeny-tiny cohort, other things start getting involved. Coach-ability and the ability to provide a unique skill-set to a particular team (i.e., "role players") will often allow for someone in the 99.9995th percentile to get the job before someone in the 99.9998th.
The number of players who make a living by being HOF-ers or GOAT-candidates is tiny, but the number of players who make a (very good) living by being really good while also fitting into a system, working well with a coach, and/or providing energy off the bench, is much much larger.
What does this mean for me?
All of these analogies are great, but let's face it. Jim Simons was still a rockstar academic by every measure (again, not even counting his creation of the most successful hedge fund in history, by far), and NBA players are, well, NBA players. I don't know about you, but none of us here at Coho are rockstars like Jim or professional athletes. Is there anything to still learn from their example?
We probably wouldn't have written this article if we felt like the answer was "no". Unsurprisingly, we feel that this way of thinking can and should be applied in almost every aspect of life. Namely, as much and as often as possible, try to find ways of doing things that uniquely fit your style, and your strengths.
Since this is a math newsletter, though, let's just focus on math-related things.
As a mathematician
At a high level, we mathematicians probably know of some of the "strengths" that exist that we could play to. Are you an "algebraist" or an "analyst"? Are you naturally better at combinatorics or differential equations? Obviously, since all fields of math either already do or probably will eventually have connections to each other, you can't be a pro at any of these without at least knowing something about the others. That said, there are obvious differences between category theorists and geometers, and you shouldn't swim against the tide if you indeed have a pull inside of you.
The types of "strengths" that we'd like to bring more attention to, however, are some of the more intangible ones.
Maybe you don't love specialization, and instead enjoy learning a decent amount about a lot of things instead of a whole lot about one really specific thing. Maybe theoretical physics is for you then, where intuition and calculation is more important than full rigor, and you need to learn everything from combinatorics and differential geometry to statistics.
Maybe you like math and find yourself looking really deep under the hood when writing your python code to try to understand exactly what your program is doing, bit by bit. Maybe it's theoretical computer science for you?
Or perhaps there are other, softer skills that you can look at. Are you good with people? Do you like managing large projects? Are you organized (something many mathematicians are not!)?
There are a lot of skills that mathematicians have that many people/organizations/companies/universities in the world need. But there are also lots of skills that the world needs that most mathematicians don't have. In fact, the curse of dimensionality all but guarantees that the following is true. There are many skills needed in the world, and so it is almost certain that most mathematicians won't have all of them. It is also almost certain that you will have some of them! We very strongly believe that it's worth your time and effort to both try to identify those, and then work tirelessly to incorporate them in ways that only you can.
The Hard Part
The hardest part about this is that you're not going to see many examples around you. That's a good sign — it means you're doing the thing. There's comfort in looking left and right and seeing others on the same path as you. And sometimes this is great. But if Al Horford (an NBA player) stopped working on his three-point shot because "no other big men are this good at threes", would he have the massively successful, long-lived, and lucrative career that he has?
Jim Simons played to his strengths — first, his strength of having "good taste" and then his strength of understanding the power of statistics in trading — and he (at least by all accounts) seems to be pretty happy with his career. On a much, much smaller scale, we here at Coho like math and we like cool T-shirts, so we decided to do both, and we're having fun doing it. What will you do that's uniquely you? Let us know!