¿Should I invest time in doing olympiad problems or should I invest time in learning more theories as a junior in mathematics?

Question

I am a junior student in mathematics interested in Automorphic Forms, but, I get the sense that in order to be a good mathematician I should spend time in doing olympiad problems (as a high school student I wasn't introduced to these type of problems.) The thing is that, I am currently coursing Ring and Modules theory, Introductory Complex Analysis and Measure Theory. I have knowledge in group theory, probability theory, real analysis, linear algebra and topology and, (as I understand, or maybe I'm wrong) in order to learn Automorphic Forms one needs lots of background in local field theory, complex analysis and harmonic analysis. Currently I don't have the opportunity to invest time in both (olympiad problems and topics in order to learn Automorphic Forms.) I was wondering if someone could advise me on what to do (an egocentric feel invades the idea that a 17-year old can solve certain problems that I can't)

Answer 1

Your question is a very relevant one in the life of someone who loves math, of someone who is young and with energy in particular as well. I believe that you don't necessarily need to do well on Olympiads in order to be a good mathematician there are countless examples throughout the story which support this statement. One example is June Huh one of the winners of the 2022 fields medal, as far as I know he started to do math a very old age (in Olympiad terms) but, he fell in love with math, and he did incredible.

Also, as someone who's participated in math competitions and Math Olympiads, I can tell you that there is a bridge, Olympiad problem - Research Problem, and is a very long bridge, but this both share many things though. An Olympiad problem may be solved in a matter of hours, days or maybe a week or 2 (depending on your math skills and experience) and they all rely on two factors Creativy-Ingenuity and Experience-Skill, as well they just focus on very similar areas and types of problems.

A research problem may be solved in years, and with help or collaboration of others, Olympiad problems are meant to be solved individually. The statements and how much do you need to know between research problems and Olympiad problems can tell you much about this difference. In conclusion, you don't really need to do well on Olympiads to be a very good mathematician if and only if you're in the right context and place.

I have to say though, the Olympiad problems are one of the best resources to enhance or learn mathematical skills and also to create a culture of problem solving with many strategies to be used in research. My advice is to continue doing your current studies since you're very old to perform super well on Olympiads since you don't have experience (if you have curiosity, you could also give it a try why not?), but try to do every once in a while, Olympiad problems, they're really worth it, you'll learn a lot.