Advantages of Mathematics competition/olympiad students in Mathematical Research

Training for competitions will help you solve competition problems - that's all. These are not the sort of problems that one typically struggles with later as a professional mathematician - for many different reasons. First, and foremost, the problems that one typically faces at research level are not problems carefully crafted so that they may be solved in certain time limits. Indeed, for problems encountered "in the wild", one often does not have any inkling whether or not they are true. So often one works simultaneously looking for counterexamples and proofs. Often solutions require discovering fundamentally new techniques - as opposed to competition problems - which typically may be solved by employing variations of methods from a standard toolbox of "tricks". Moreover, there is no artificial time limit constraint on solving problems in the wild. Some research level problems require years of work and immense persistence (e.g. Wiles proof of FLT). Those are typically not skills that can be measured by competitions. While competitions might be used to encourage students, they should never be used to discourage them.

There is a great diversity among mathematicians. Some are prolific problem solvers (e.g. Erdos) and others are grand theory builders (e.g. Grothendieck). Most are somewhere between these extremes. All can make significant, surprising contributions to mathematics. History is a good teacher here. One can learn from the masters not only from their mathematics, but also from the way that they learned their mathematics. You will find much interesting advice in the (auto-)biographies of eminent mathematicians. Time spent perusing such may prove much more rewarding later in your career than time spent learning yet another competition trick. Strive to aim for a proper balance of specialization and generalization in your studies.

I would say that olympiads build some, but far from all, of the skills needed to excel at mathematical research. I'd compare it to running 100 meters versus playing soccer. Usain Bolt is probably a better soccer player than the vast majority of the population, because he could outsprint anyone and because he's generally in fantastic shape. But that doesn't mean he's going to be able to play on a professional team.

Being a successful researcher requires

• the ability to learn new fields of mathematics, and develop ways of thinking about them that others haven't.
• the discipline to spend months or years returning to a problem and trying new angles on it.
• (or at least is strongly aided by) the ability to communicate and "sell" one's results, in writing and in talks.
• the ability to write good definitions, that will be useful and cover the boundary cases correctly.
• the ability to form an intelligent guess as to which unproven statements are true and which are false.
• the ability to hold a complex argument in one's head and play with it.
• (or at least is strongly aided by) the ability to find clever technical arguments.

I would say that olympiads are very helpful in developing the last skill, somewhat helpful in developing the fifth and sixth, and not at all in developing the first four.

I definitely, at some points in my research, find myself needing lemmatta which would be fair to put on an IMO or a Putnam exam. And when that I happens I feel myself relaxing, because I know I can do that. But I also spend a lot of my time trying to learn how to think about a subject, or figuring out what to prove, or trying to figure out how broadly a phenomenon holds. And those are not skills which I found olympiad training helpful in.

In case someone wants to know my Olympiad credentials to evaluate this advice, I was the first alternate to the US team in 1998 and, during my senior year of high school, I regularly came in somewhere in the top 10 spots in national (USA) contests.

Keep in mind that what you have here is a correlation, not a causation. While doubtlessly Olympiad training would help develop some skills necessary for research, I think it is likely that many of the strongest mathematicians participate in these competitions when they are in high school and go on to do research later.

I can say from personal experience that the bulk of people who train for the IMO tend not to become any more exceptional at research than any other person/s who take up the subject - with intent to become a researcher - at university and beyond.

Some of the top-performing students at the IMO - including a good number of Gold and Silver medalists (from the US atleast) - I have known: and I can't say that they became any more exceptional at research than any other non-IMO participants and / or top-scorers.

Basically, the competition tends to make participants into very sharp-minded and 'clever' problem solvers (which, perhaps, has some advantages in some contexts in research); but as far as giving you a -significant- advantage, it really doesn't do much as far as I've seen.

Rather, follow the advice of a well-known mathematician, John Milnor -- think carefully, think deeply, and work patiently and diligently at whatever problem you're working on.

I think you might find that proves the best approach to research, regardless of academic specialization.

cheers

Students who manage to make IMO are much smarter than the average math major. People will disagree with me, but anyone who's gone through undergad with IMO medalists will know that I have a point. To make it to the top levels of any competitive intellectual pursuit, competitors must already have a very high level of base intelligence that's transferrable to many other domains. Every IMO participant I know is insanely smart in general, not just in math competitions.

I don't buy these two common arguments:

• "IMO students have the same mathematical abilities as any other math student."

This simply isn't true. It's no coincidence that more than half of the participants of the US team get their PhD's, and essentially every US IMO participant who chooses to pursue graduate school ends up getting into a top program and suceeding. Do some Google searching for past participant's names if you don't believe me. It's also not a coincidence that a very significant percentage of IMO medalists end up taking graduate math courses their freshman year of college. That's much more impressive than the average undergraduate math major.

Fields medals and the like are indicators of extreme outliers, and should not be the main factor used when guaging the general research ability of a population.

• "IMO students only train on learning competition tricks."

IMO performance doesn't happen in a vacuum. You have to consider the kinds of kids who win IMO: Smart, motivated, and passionate about math. Many IMO medalists will learn far ahead in high school, and many will learn enough math to conduct legitimate research during high school or early undergrad. This gives them a rather significant head start.