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I am currently applying to PhD programs in mathematics and need to complete a purpose statement as part of the application. There are several things I'd appreciate any advice you can give me on the following:

  1. What is a good way to end the letter in a strong way? I wanted to say something along the lines of "I greatly look forward to further understanding the intricate theory of XYZ" but this seems a bit cliché.
  2. I wanted to mention my specific interest in school XYZ/professor XYZ, but I don't know enough about his/her research to say something intelligent or insightful about it. What should I do? Is it better to have some indication I did my research for this school than to say nothing at all? To be clear, I am interested in the field of these professors but my knowledge is too shallow to fully understand the impact of their work. How can I add something substantive but not sound like I'm making things up to sound smart?
  3. For one of the schools I am applying to, a big reason I want to go there is due to their well funded program. Should I mention this or will it seem tacky?
  4. What is a "smooth" way of mentioning I want to work with a professor? It seems too abrupt to just say "I want to work with professor XYZ". What if this person doesn't have any current students (or at least doesn't have them listed in his/her Web page). Should I take it that this person is not currently taking students?

Any and all advice would be helpful. Thanks in advance!

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What is a good way to end the letter in a strong way?

It doesn't really matter too much as long as it's not too cliched (although even that's probably fine) or overly memorable. An example is to say something nice like "Given my background and the excellent research faculty at University of State, I look forward to making a contribution to the research program at University of State."

What should I do? Is it better to have some indication I did my research for this school than to say nothing at all?

It's definitely better to have some indication that you did your homework. You don't have to have any particularly insightful things to say, just explain why you found their work interesting and why you want to work with them. Those are questions you should answer for yourself anyways! Definitely don't make things up to sound smart.

For one of the schools I am applying to, a big reason I want to go there is due to their well funded program. Should I mention this or will it seem tacky?

Mention it tactfully. Say something like you appreciate the commitment they make to their graduate students and that it speaks to their commitment to good scholarship.

What is a "smooth" way of mentioning I want to work with a professor? ... What if this person doesn't have any current students.

There's nothing wrong with mentioning that you want to work with specific faculty; it's expected that you have some ideas of what you want to work on before you get in. So say who you want to work with, but make sure you list several faculty members who work on things you're interested in. This way, should it happen that the person is not taking students, your application will still get a fair shake.

This next piece of advice is maybe controversial given how many emails professors get, but after you have applied, you can send a short email to the faculty you mention in your statement of purpose telling them that you've applied, you are interested in their work, and if they're looking for students to keep your application in mind.

Should I take it that this person is not currently taking students?

Not necessarily but it is strange for an active research professor to have no students. Ask a graduate student in the department why they have no current students.

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    "[I]t is strange for an active research professor to have no students." I wonder what field you're in. In mathematics, this is quite common. I just looked at my own department and roughly half the research active faculty currently have students. Nor is having students perfectly correlated with research quality: one of the greatest living mathematicians has had three students (two were coadvised) over a 60 year career: genealogy.math.ndsu.nodak.edu/id.php?id=34222&fChrono=1. – Pete L. Clark Dec 4 '15 at 23:28
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    By the way, I don't mean to be harsh: the rest of your answer is completely on point, and I upvoted it. – Pete L. Clark Dec 4 '15 at 23:31
  • That's a fair point, and I agree in pure mathematics it's not a big deal to only have a handful of students (in fact it might be the norm as you say). It's slightly different in applied math where I come from. Thanks for pointing this out. – Sumedh Joshi Dec 4 '15 at 23:37
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    I didn't say pure math. FYI, the percentage of applied mathematicians in my department who have students is slightly lower than the departmental average. FWIW, my guess is that globally the trend would be the opposite, but I could certainly find hundreds of tenured applied mathematicians with no graduate students. – Pete L. Clark Dec 4 '15 at 23:42
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    @SumedhJoshi Thank you for the response. My intention with question 2. is to ask what to do if it is difficult to come up with why something is interesting. A lot of times in mathematics, it takes a lot of background knowledge before one can appreciate (e.g. find interesting) a result. I guess I can say that I am interested by the research because it involves topics/objects I am familiar with and would like to study, but short of this I find myself stumped at saying something "substantive". – Kevin Sheng Dec 5 '15 at 1:24
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Tips*

  1. I would NOT follow rubric guidelines that are plastered all over Father Google. Advisors, committee pupils etc that are reviewing your app amongst a thousand others will see the same "This got me into Yale so you should do it to" ;goes straight to the trash.
  2. What I would do is make sure I answer all necessary criteria they have with no glossing/rambling and make it sound meaningful. Definitely follow proper grammar structure, rules etc but make sure that you standout
    You'll know it's ready when you read it and you're like I wrote that? :)

*I didn't answer your questions directly because our input on how to write what will sound like someone else wrote it. Needs to be consistent!

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  1. You don't need a strong ending because this isn't an article, and you want to keep your essay concise.

  2. Mention the field or sub-field you are interested in. The department will know who is working in it. Don't worry about sounding pretentious -- this essay is not going to be published anywhere -- but do make sure you don't make statements you really don't know enough to make with confidence.

  3. Don't mention the financial support available at the particular school. However, since many supported students at UW and many other schools are expected to serve as teaching assistants, it might be helpful to say something about your experience and interest in teaching.

  4. See Q2; also, don't assume the person is not taking students.

I would also add that the link @FirebladeDan found looks very helpful. And it's a model of conciseness!

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