I am currently applying to graduate schools for a Ph.D in math. One of the schools to which I am applying has a section in which I can upload a writing sample. The description says "Please upload a sample of written work that is relevant to your application (e.g. seminar paper, report, magazine/newspaper article, book excerpt)." Unfortunately, I am not published, but I have written several expository papers on relatively advanced topics for different classes. For example, I wrote a ten page paper on the Riemann-Roch theorem for an algebraic geometry class. Is that the sort of thing I should submit in lieu of a published paper, or should I leave the section empty in my application?
In a similar vein, there is a section titled Publications and Presentations where I can submit material. The description for this says "Provide a list of publications and/or presentations related to your academic or professional background (include the PubMed ID if applicable)." I gave a presentation on my research (the triple bubble problem) at an undergraduate research conference a few years ago. Should I submit the slides for the presentation under this section, or should I leave it empty as well?
I know that math is sort of an interesting case for undergraduates in terms of original research, seeing that it takes quite a while to really start doing meaningful research. I didn't know if I should consider this when thinking about whether or not to submit presentations/papers that are by no means completely original ideas, but are more of the expository nature. Thanks for the help!