I’m applying for math PhD programs this year. I’m a junior at UCB, and I’ve taken standard grad math courses such as algebraic geometry in the last year. This year I’m taking grad seminars and topics courses, and the topics covered this semester include geometric rep theory, symplectic field theory, TQFT and microlocal sheaf theory. I have no time to do any meaningful research on them until the December, and I have no previous research experience. However, I’ll have given ten expository talks related to them since the last year until the end of this semester. I made a slide for one of them, but I have nothing to submit other than that, as I used blackboard in most talks.

To complement my lack of research experience, should I write an expository paper on a topic of my interest? Or is there anything else more effective for admission? I can write an expository paper within this semester, but I personally prefer studying recent topics to be up-to-date than writing such papers, which is often time-consuming.