To the OP, unless they are changing the common app, the place to put summer activities is the same place to put extracurricular activities in general. There is no separate “summer” section.
I understand the worry - especially if you yourself feel that you are one dimensional, but if that is truly the case, do you really want to be at a place that emphasizes diversity that much?
Where are you considering applying? If you are concerned that you don’t have a great chance for HYPS, so what?
You can only do your best and be true to yourself - many if not most highly selective colleges will have spots for kids who have figured out what they want to do - just as they have spots for kids who still have to find their way, as they have been good at everything they’ve ever done.
I have no doubt that there is a great college future ahead of you, and you will find it your way, by doing what your math training has taught you - analyze and find solutions.
Before long, you will have multiple acceptances in front of you, and you will have to use another math skill - prove it to yourself.
@Erin’s Dad and @doschicos are right. There is no dedicated “What did you do during your summers” section on the common app. I’m in the camp of follow your passion. You obviously love math, enjoy it and want to find ways to pursue it. (Canada-USA MathCamp signals that strongly).
Don’t twist yourself trying to make you look like some “ideal” candidate. Be yourself. Really. I think colleges want that type of student, even very pointy ones like you, over the student who follows a recipe but not their interests, to gain admittance.
I would just look for different avenues, instead of just solitary pursuits - try tutoring younger students (this could continue into the fall and winter); see if you can find a community of sorts for your interests… For example, take a look at the website, Expii. It’s a wikepedia for math geeks from what I can gather; the site is still in its infancy. Personally, I like the Feynman Lecture idea but again, see if you can find a community of sorts (even one other person).
I am confident that a motivated young person such as you appear to be in your posts will find a college that will appreciate your commitment to learning quantum mechanics and other math topics. I think you are more worried about how your application will appear than is strictly necessary. Follow your passion and interests, and there will be a good school out there that will welcome you on to their campus. The summer stuff will be part of the overall picture that you paint which will be reflected in your application. If you have other things you want to do that will contribute to being multi-dimensional that is fine, but I do not think you should add more activities simply to diversify for application purposes solely.
You aren’t too late yet. Ask around and look for internships asap- you don’t want to be self-studying when you could be doing something that is golden to write about in essays.
btw most summer programs/community service aren’t looked upon very favorably. And its by no means too late to do research in a lab.
How much programming do you already know? That’s a concrete skill that you could pick up and grow with online tutorials over the summer. Even pure math majors and theoretical physicists need to do some programming. Matlab and/or Python would be useful in the long run.
The colleges you mention reject tens of thousands of top candidates each year. So your fear that you might be rejected (post #15) is legitimate, and the way to diversify is by applying to a range of top schools beyond HYPSM.
As for your summer, IMO colleges like to see students that have a genuine passion, and that have taken that passion as far as they possibly can.
You know what you’ll need to show and prove in your application to top schools if you present with heavy math including a summer of self-study? That people like you and you like them and find it easy to engage with other humans. Does/Can your app show that? If not, THAT needs to be your big concern. 2 cents.
@dyiu13 , I think you summed it up perfectly here. My son’s HS math teacher told us “There is sometimes a stigma against severely introverted math geeks like us, so we have to work harder at the other areas of life that don’t come so naturally” This advice meant a lot to my son.
OP, for a mathematician to feel a little self-doubt (you didn’t say this, I am projecting it) or to sense confusion with the college admit process is not at all uncommon. An analytical mind wants to know all the variables used in the decision process, but many times the process is much more complex than can be describe, and includes many more factors beyond our own control. After all that, it can come down to as much art as science, which can be hard to accept.
Don’t fall into the trap of putting yourself completely into a box. You haven’t responded to the questions about what other EC’s, if any, you have during the school year, or what other schools you might be considering.
I think it’s important. But if it’s too late for you then trying to do something. Get a job even. Don’t sit idle.