How to make sure graduate admissions committees are able to recognize/verify the material a student has self studied
Letters of recommendation which attest to this, and also perhaps the math GRE. You can't believe how much garbage I see in personal statements about how much students have "mastered" a particular mathematical topic studying on their own, only to find out later they had "forgotten" what they had learned months ago after I meet them in the autumn and they're banging on my door asking me to take them on as their potential advisor. So this is why I trust letters of recommendation over the word of the student by and large these days. Please get one of your letter writers to attest to your self-study. Generally, the more senior and influential the letter writer who says this, the more likely I will believe you've self-studied what you say you've self-studied.