I self-studied a bunch of university-level math. How can I demonstrate this to colleges and use this to strengthen my college applications?
Ideally, do all of the following.
Contact local universities/professors to see if they can help you enroll in the corresponding courses and get it on your transcript.
I’ve heard of cases where a high school student was allowed to take a university course for free under the agreement that they would apply to the university. Make sure to get an A in the course, impress the professor, and ask for a letter of rec. (Impress not just with your grade, but also with your attendance and participation, which should include insightful questions & discussions during office hours.)
Create a personal website (advice here), apply the advanced math you learned to projects, and showcase those projects on your website.
Project examples include but are not limited to science/technology/engineering fair projects, writing about your own mathematical investigations, creating your own educational resources that others can learn from.
When you display a project, it’s important to give a comprehensive overview with pictures, graphs, and code snippets or links to code (if applicable), as well as an explanation of why you did the project, any challenges you encountered, things you learned, etc.
A regular school transcript won’t provide enough information for college admissions officers to know your educational background – so your portfolio needs to make it obvious that that you did a bunch of super-advanced stuff, you can explain it and talk coherently about it in general, and you have a high degree of passion and ownership in what you did.
If you find the opportunity to work with a mentor on any project, work extremely hard on that project in particular and ask for a letter of rec.
Work it into your essays.
What were your motivations? What challenges did you face and how did you overcome them? How has this influenced your life goals? Don’t write every essay about math (otherwise you’ll seem one-dimensional), but definitely make sure to highlight it.