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I know this is a topic frequently talked about here, but I'm coming from a different point than any of the posts I've found so I figured I'd ask.

This post is less of a "can I do it?" question and more of a "what do I have to do to do it?".

Currently I'm finishing up my 3rd year of high school, but graduating a year early because my school doesn't have any more courses along my paths of interest (Math, Physics, Comp Sci). While applying for colleges, I decided to go to UTSA because of some meetings I had with a few of the professors there who said they'd be glad to mentor me into mathematics and that there would be plenty of research opportunities (one of the main focus's of their honors college), and while I know it's not a top 20 school, I really enjoyed my experience there. I'll be entering taking Calc 3, Linear Algebra, and a Proofs course because the school wouldn't let me skip any more than that on my first year before I prove that I can handle it.

Over this last school year I've taught myself Calc 1/2/3, Linear Algebra, and basic proofs and set theory. (Apostol Calc 1/2, Spivak Calc, Velleman Proofs, Greub/Werner Linear Algebra) and want to continue self studying (under supervision of a prof hopefully) while taking my courses. If I spent a solid 6-8 hours a day basically everyday working towards progressing so I can take graduate courses as soon as I can, do you think this would be enough to be able to weasel into a good graduate program assuming I have good test scores, GPA, as much research as I'm able to help with, and participating in the campus math clubs.

Basically, I'm wondering how much I have to make up for a lackluster high school career. During middle school I was near the top of the state in UIL and Mathcouts consistently, but my high school did not have any academics math teams so I missed out on all that.

Another possibility would be to transfer to UT Austin after my first year to their mathematics program, but I'm not so sure that would be beneficial.

Ultimately, academia is the dream, so I'm willing to dedicate myself to seeing it happen if it's possible.

Thank you for your help!

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    Results at school make zero difference to PhD applications. I doubt if there's even a section on the application form for them. Also, there's no prize for getting into a PhD as soon as possible. Take your time and develop your mathematical maturity.
    – astronat supports the strike
    May 13, 2018 at 19:47
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    @cactus_pardner Thanks, I'll edit that right now
    – Quinn Murphey
    May 13, 2018 at 20:04
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    @astronat For me, it's less of my high school results affecting my chances as it is not being as far along as I could/should be. Many of the top programs are highly selective. Also I'm not trying to rush through my courses, just trying to open up opportunities quicker so I have things to put on my resume such as quality research. I still value comprehension over speed due to my innate love of understanding why and how in math, but realize the necessity for some speed.
    – Quinn Murphey
    May 13, 2018 at 20:08
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    @cactus friendly rejoinder: I'm trying to reduce the USA-centricity of this site one comment at a time.
    – astronat supports the strike
    May 13, 2018 at 20:11
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    If I were you I'd have stayed one more year in HS for a better undergrad institution...Some people say otherwise but unfortunately names do matter
    – xuq01
    May 13, 2018 at 21:39

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It's too early to stress about grad schools in your situation. But let me share a few observations I have made being a student at a top 10 Math PhD program in the U.S

  1. In every incoming class of last 3 years, there were a number of American students that came from a no-name schools such as a tiny liberal arts college nobody heard or a low ranked stated university. This means that if you are the best student in your university and make good friends with your professors, there is a decent chance to go to a much better school. I specified American students because international students pretty much all come from premier institutions.
  2. Do an REU in your sophomore or junior summer. The point of REU is not to prove something new, but to expose you to the research world. Ideally, impress your REU supervisor well enough to ask for a recommendation letter from him/her.
  3. Do not underestimate the Math GRE. Take your time to practice and get 800 minimum - 760 at the very least.
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    Thanks for the advice, I know I’m stressing too soon, and we’ll see if I’m cut out for it during my undergrad, but it helps to know that not everyone at the top 10 comes from a top 10
    – Quinn Murphey
    May 14, 2018 at 0:54

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