I'm applying for graduate programs in pure math. I'm not too sure what to write for a statement of purpose. I am very interested in functional analysis and I have done an REU work that involved with that subject and is being submitted for publication. I figure I should include those, but I'm not sure how long to make it or what to actually put in the essay. Do I talk about what classes I have taken? Future goals? Research interests? Should I mention specific faculty members? Thanks for any advice!