It seems as if I'll be taking the next year off and studying a lot of mathematics on my own and apply to graduate schools in the next cycle. There's a lot of advice both on this website and Academia StackExchange for such students. However, I'd like to know how can students who have studied material outside of class better convince the admissions committee of the work they have done thus far. I know it's generally difficult to do this as an international student; the situation must be different for U.S citizens, I suppose.
Assuming I don't get to take courses which I'll be self studying over the next few months, how should I go about making sure the admissions committee take due notice of the work I have done, and more importantly, they can recognize it, if not verify it completely. Of course, I will keep in touch with my math adviser. In addition, what else can I do? Say, I am studying abstract algebra on my own in the summer, working out problems in, say, Dummit and Foote's book. Should I, perhaps, make a website (a Google site), periodically type the solutions to the work I have done and post it online, so when I apply to graduate schools, I can refer admissions committees to this portfolio of sorts.
In a nutshell, I'd like to know, especially for students who have gone through such a process/situation, of the list of best possible set of actions one can do to make sure the work one has done outside of class is duly considered, if someone who is interested in mathematics schools doesn't have a large number of math courses.
For more information on my situation, please see this link. Feel free to weigh on any website.
Write about it in your personal statement. Your personal statement for graduate school is your handle to explain yourself beyond what is on your transcript. It's the best vehicle available to you to explain how your official transcript undersells you as a candidate.
Personal anecdote: I had to withdraw from a mathematics course due to health complications, and spent a paragraph talking about that course and how much I enjoyed going to lectures and how I was sad that I had to withdraw but was excited that I was able to do research with the professor anyways that summer.
The most important part of the narrative of the personal statement is to use it to explain and preemptive defend yourself. It takes a string of As and Bs and explains how they fit into your life and what else was going on while you did it.
In an ideal world, you'd also get a professor to testify to this. I don't know if it's too late for this, but if you picked a professor who was going to write you a letter of req and kept them updated with the problems you've been solving and ideas you've been learning, the fact that that professor would be able to confirm the veracity of your narrative in their letters of recommendation would also be helpful. Other ways of demonstrating such grasp of material, such as writing expository papers or working a job that requires such knowledge would also advance and confirm your case.