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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.

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    Anything that conveys the idea that you have understood the above-mentioned topics is probably highly helpful to your PhD admission. Just make sure that your paper actually shows competence (i.e., avoid oversimplifications, dumbing down and stupid mistakes). – darij grinberg Sep 11 '16 at 5:03
  • Thanks for your advice. I will be careful about the points you made. – AK-47 Sep 11 '16 at 5:12
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    Future students reading this might be able to, indirectly, extract some advice about the relative priority of "advanced classes" and "research experience"... – Nate Eldredge Sep 11 '16 at 5:39
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    A side note: if some of your professors attended your talks, maybe they could attest to your understanding of the topics and the quality of your exposition if you ask them for a letter. – coldnumber Sep 12 '16 at 4:59
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    Do you have a professor in your department that you could discuss this idea with? They would be able to tell you a lot more than we can about whether your proposed topic is interesting, could give suggestions, and obviously can otherwise give you good insight into how you might prepare for graduate applications. – Patrick Sanan Oct 26 '16 at 16:00
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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.

  • I apologize for a late reply (In fact, the deadline for applications is today). It turned out that two of the professors in my talks wrote a LoR for me, and I wrote several very brief expository notes, one of which I submitted to some of the programs. Only two of top math programs I applied to provided an option of uploading supplemental materials such as theses and notes, so as some of the people here implied, even top programs aren't particularly concerned with checking applicant's papers. Thus, your second paragraph is very accurate to the reality. – AK-47 Dec 16 '16 at 5:19
  • I didn't have enough time to consult with the professors about the notes, and notes I made were rather brief. I later learned that some of them happened to look like a mere subset of notes made by master's students for completion of degree. I realized that, as Kimball said below, learning things are more important than spending a lot of time for this. Fortunately, I didn't spend a lot of time, and I learned a lot this semester! – AK-47 Dec 16 '16 at 5:32
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I agree with points made in other answers, but want to point out: if you're at Berkeley (I assume that's the UCB you mean) taking loads of grad classes and doing well, then you should get good letters and get into a top program without much trouble. Doing research isn't necessary for US PhD programs, and as discussed elsewhere on this site probably won't help all that much under normal circumstances anyway.

Also, if you haven't written papers before, you probably don't realize how time consuming it really is (at least to do a good job of it, and avoid mistakes---if it's poorly written, it might hurt your chances) and may take away from your other studies.

In summary, since you want to spend your time learning other things, I think you're better of doing that.

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Writing a good expository paper for a journal is by no means simple, even if you are prepared by having given ten talks on the topic. A journal would want to see that you can give the big picture of the cutting edge research, and usually not an overview of what most people know already. So I would say that writing an expository journal paper at such an early career stage is as hard and time consuming as doing actual research.

As an alternative, you may consider to write them as lecture notes. As such they would follow more closely the lectures you gave and would have grad students as target readers instead of researchers. The downside is, that you could not regularly publish them in a journ, but could "only" host them on your homepage, your blog or a preprint server.

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