Is writing an expository paper helpful for math PhD admission? 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.
An expository paper is useful in the abstract, but how much it will actually help you depends on the circumstances.

The basic problem is that if you hand a random mathematician an expository paper on microlocal sheaf theory, they won't know what to make of it.  Is it even correct?  Is it plagiarized?  Is it completely routine (closely following standard sources), or does it show creativity in the organization or explanations?  Unless the reader has real expertise, they won't be able to judge it without a lot of time and effort, and there's a real chance that nobody on the admissions committee will have that expertise.  If everything goes well, they'll forward your paper to an expert who will give them an informed judgment.  But if that expert turns out to be busy or disorganized, your paper may fall through the cracks and end up not counting for much with the admissions committee.

You can get around this by making sure at least one of your letters of recommendation discusses the paper.  If the recommender can vouch for the quality and originality of the paper, then that will mean a lot.  Of course this means you have to convince someone to take a serious look at it, but your chances of doing this are at least as high as the admissions committee's.

If you are writing the paper under the supervision of a faculty member who is happy to read it and write a letter of recommendation, then that's great.  I'm not sure I'd recommend just writing one on your own, however.  It could work out well, but there's a risk of not getting much admissions benefit from it, so this would make sense only if you feel writing the paper would be a rewarding activity even aside from graduate admissions.

If you've already given talks that will be discussed in your letters, then that could be good enough.  An expository paper comes across as more serious and convincing than just some talks, but not necessarily enough so to justify the extra work.