I'm currently an undergrad and I want to get into an academic job for pure mathematics because I want to do both research and teaching in the field. Particularly, I'm interested in a field related to Algebra. Yet I have undergraduate friends who seem to be super dedicated and passionate in pure math as well. For example, I have a friend, and she clocks in long hours of doing math to the point of being similar to a full-time job. She seems to have received plenty of exposure to math even before she pursued her studies in uni.
My background in mathematics may be different from many other math majors, as I've been exposed to writing formal, rigorous proofs and doing thought-provoking math problems (similar to those seen from competitions), so I have experience in problem-solving, before pursuing uni. Then again, I have this feeling that all the other pure math majors are better than I am, which is what makes me feel like a moron who is currently questioning my self-worth.
I appreciate being with like-minded people, who all love mathematics just like I do, but nevertheless, I can't shake off this internal fear that they will go to top-ranking grad schools and subsequently take my job while I face the possibility of failure to break into academe. I guess this is me being ultracompetitive but it seems that the academic job market is a nightmare (and I can't see myself doing applied math in my life). Nonetheless, I find that this mindset will most likely damage relationships that I have with my pure math friends, as well as my mental and emotional well-being.
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