Ways to display extra-curricular/independent learning I am a mathematics student and have been extensively reading textbooks cover-to-cover in my own time.  Thus, my transcript does not say that I have (for example) taken differential geometry, even though I know diffgeo as well, if not better, than if I took the course.

How would you suggest that I display such efforts so that grad schools can be aware that I am more knowledgeable than my transcript might suggest?

Thanks!
I faced this exact problem when I was applying to grad schools. Here are three ways that worked for me (in order of importance):


Include this information in your statement of purpose. 
I do not recommend going through a laundry list of all the books you have read, however, I would mention the new topics that you are familiar with, how you have been learning them (independent study), and how you are looking to continue furthering your knowledge (which is presumably by study at the university you’re applying to). 
Have your letter writers vouch for your independent study.
Most universities require letters of recommendation as part of your application. It is a good idea to choose letter writers who can speak to your different strengths, and I think it would be beneficial to have at least one of them who is familiar with all of your self-study. I met with my professors who agreed to write letters for me so that we could talk about my application and preparation for grad school. One professor asked me specifically to prepare a list of topics I’d studied independently so that he could write about it in his letter to the university.
Contact professors at the schools where you want to apply.
Getting professors familiar with your background can be helpful so that they can point you to even more resources for self-study, and they may also be helpful when going through the application review process. There is no guarantee this will influence your application, however, it possibly has some bearing on it, and it more importantly will help you become better at self-study regardless.