In last week's newsletter we wrote about the jump from doing "homework" to doing "problem sets." This is a jump that typically occurs towards the end of one's high school experience or at the beginning of one's college experience, and will last either until mathematical retirement or until one starts embarking on full-blown research. It is only fitting, therefore, to write about this next jump — from problem sets to research.
This will be a multi-part topic, though, because there is a lot that goes in to this jump. Even with multiple parts, we still probably won't cover everything, but that leaves a lot of the fun for you to discover! (Also, this article will assume we're talking about math research, but the same can all be said for physics, computer science, engineering, etc.)
A Humbling Experience
So here you are, a third year undergraduate or maybe a first year graduate student and you have done a lot of problem sets in your life. You're learning some pretty tough fields and you're comfortable solving some pretty hard problems. In other words, you're feeling pretty good :)
You therefore decide to start exploring the world of research, thinking that it's time to start specializing and burrowing your way to the frontier of knowledge. Nice.
You decide to go to some seminars and/or download some recent papers in cool-sounding fields and something weird happens: you don't understand a single word of what anyone is saying. You don't even understand the titles of papers or talks. You literally haven't even seen most of those words before. Time to panic? No (though you probably will on occasion anyway). Time to get to work? Yes.
The NBA of Math
(Yes, I'm going to talk more about the NBA in this article.)
What does the NBA have to do with any of this? A lot, actually.
Once every 10 years or so there's an NBA team that is really, really bad. Like, "loses 90% of their games" level of bad. Let's call this NBA team "NBA Team Z". Every time this happens, there are talking heads out there who say "College Team X could beat NBA Team Z", for some powerhouse college X. And they would be fantastically wrong in this statement. Here's why.
A college team is made up of largely 19-21 year old kids who are really, really good at basketball relative to other 19-21 year old kids. Let's say these kids started playing basketball (with any level of seriousness) at the age of 8. Being kids, there's all kinds of kid-related responsibilities that they still have: going to school, having crushes, and just generally being kids.
Let's therefore say, generously, that they can average 2 hours of real basketball practice a day. That's 12 years at 2 hours a day (no vacations, nothing) and we get to 8,760 total hours of practice. Keep in mind that this is practice against other kids, and with coaches that are likely drawn from some pretty small geographic area, being paid not-that-much.
Now consider someone who has been in the NBA for even just 5 years. By the time you reach the NBA, basketball is your full-time job. The court and the gym are your offices. Let's therefore say, conservatively, that an NBA player practices for an average of 6 hours a day. In 5 years, this is almost 11,000 hours.
This to say that a mere 5-year veteran in the NBA has played in the NBA for longer than most 20 year-olds have played basketball, at all. Moreover, this NBA-training is against the world's best competition, with the world's greatest coaches, trainers, and nutritionists.
In other words, even the worst NBA players are much, much better than the best college players. It's not even close.
Now Back To Math
The above reasoning about NBA players is directly related to what you're going through when you first jump in to research-level math. Until this point, you've been "taking classes." A class has a specific subject that it focuses on, using pedagogical tools like textbooks and (hopefully well-crafted) lectures designed specifically to expand your toolbox. The communication is designed for students.
And, as a student, you have other student-related responsibilities. You have other subjects to study (even while majoring in something at university), friends to make and hang out with, etc. And that's great, don't skip that stuff.
But when you look at a tenured professor of math (or computer science, or physics, or whatever), you have to understand that this person has been doing this full-time for quite possibly longer than you've been alive for. Moreover, this person has been collaborating with, and learning from, the very best in the world, for years. While you've been reading textbooks, this professor has been learning directly from, and solving problems with, the person who wrote that textbook.
And finally, when you go to these peoples' seminars and/or read their papers, they're no longer communicating to you, the student. They're communicating to the other world experts in the room. They're playing a totally different game.
Don't Be Discouraged
All of this is not meant to discourage you — quite the opposite, actually. It's meant to show you what's possible.
We all already know (hopefully) just how much friggin' better at basketball NBA players are than the rest of us. And most people also know just how much better they are than even a very talented college player.
However, we (society as a whole) tend to talk a whole lot less about just how friggin' much better at math a professional mathematician is (or insert your favorite technical field here). And just as an NBA player is much better than even a very talented college player, so too is a professional mathematician relative to a talented college student.
Of course, there are your Caitlin Clarks and Lebron Jameses who come straight from high school or college and immediately succeed amongst the pros. Similarly, there are your Ed Wittens and Terrence Taos who do great things at young ages. To be honest, you're probably not one of them, and that's totally okay.
What's less okay — and what we want to help you avoid with this article — is the all-too-common story of a 20-year-old college rockstar athlete having no idea what awaits them at the next level. They think they're hot s*** and refuse to work hard, only to be relegated to some obscure 2nd-tier European league (or worse, an accounting job (no offense to any Europeans or accountants!)).
So, if you're standing on the brink of running with the big dogs in the world of cutting-edge research, congratulations to you! It won't be easy, especially at first, but that's because you're leveling up. Don't run from it, embrace it. Put your humble-hat back on and start that many-years-long process of learning from the best so that one day you, too, can be counted as one of the best.
And then be sure to be nice to the next generation of folks trying to do the same!