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I intend, in the somewhat near future, to engage preparing my graduate school applications for next year. I have worked hard to secure a solid application as far as coursework, grades, recommendations, etc., etc., though the statement of my research plan is very important to me (and, as a friend of mine who is on an admissions committee at a top school has informed me, it is far more important than most students believe it to be).

However, I find myself at an impasse; I have research interests which lie at the intersection of a broad array of wider mathematical disciplines (algebraic geometry, algebraic topology, a bit of number theory, representation theory, categorical algebra, and even model theory--I always try utilizing my mathematical toolkit in assessing problems in mathematical physics, as well).

If I just go on in my application listing these disciplines, I won't be taken seriously. Though if I am too particular, I risk appearing too specialized for the research being conducted at school X (and I am broadly interested, though this can be a boon if not taken too far).

So rather than expressing my interests (and potential interests) in the following way:

--algebraic geometry --algebraic topology --arithmetic geometry/algebraic number theory --n-categories/topoi --representation theory --etc., etc.

I would like say something like --motivic cohomology, etale homotopy, Hodge theory, stacks, D-modules (algebraic geometry/topology) --braided monoids and algebras (representation theory, category theory) --model-theoretic proofs of Mordell-Lang and geometric stability theory (model theory/arithmetic geometry) --n-categories, higher constructions with topoi (this ties in with my interest in etale homotopy).

What would be a good strategy here? I don't want to seem unfocused or naive, but I don't want to leave out any of the many things in which I have some degree of interest? (Many of the subjects listed here are things which I have actively pursued outside of the classroom to some degree, some of them at an advanced level--e.g., motivic cohomology and etale homotopy).

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Which country and what sort of institutions are you applying to? –  Bugs Bunny Jun 4 '10 at 10:54
    
In order of interest, and (roughly) reverse order of likelihood of admittance: Columbia, Chicago, Michigan, Penn, Rutgers, Purdue, SUNY Stony Brook, Toronto (Masters), UBC (Masters), ALGANT (EU Masters), UMass Amherst, Notre Dame, Utah, CUNY, Oregon, Arizona, Michigan State, LSU, UConn, Albany, and a few others. I feel as though my chances are decent; I have a lot of advanced research, good grades in graduate-level coursework, and a high subject score. However, I did do poorly in some elementary courses (calculus) when I was young and immature, though I hope that my GRE will remedy this. –  lambdafunctor Jun 6 '10 at 6:18
    
Best of luck!! How did you manage to satisfy religious requirements to both Notre Dame and Utah:-)) ? It may be a good idea to e-mail a shortened version of your statement directly to a person with whom you would like to work in each particular institution. Just imagine him/er leaving next year or being on non-talking terms with director of graduate studies... –  Bugs Bunny Jun 10 '10 at 6:50
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4 Answers 4

Some random thoughts:

1) I recommend that you discuss this with a professor who knows you well and show him drafts of your statement.

2) There is no reason why you need to submit the same research statement to every school. You can focus on different research topics, depending on the strengths of each department.

3) Your list of interests above is way too long and broad. I doubt it will be taken seriously. Focus on only one or two and discuss them in enough depth to show that you really know more than just the terminology.

4) I agree with everybody else that the statement should be no longer than two pages, no matter what.

5) Nobody expects an undergraduate to have much breadth or depth in their knowledge of mathematics. What you want to demonstrate is your desire and commitment to building greater depth in your knowledge of mathematics. Although you don't want to appear too narrow (and this does not seem to be a problem for you anyway), demonstrating breadth or an interest in breadth is far less important than showing the desire for depth.

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+1 for thought 1). I still shudder when I think about the awful personal/research statement I was originally going to send out with my applications. –  stankewicz Jun 4 '10 at 21:34
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The "research statement" is a good idea, and usually serves two purposes.

(1) classifying applicants. You see, if the department is large, you want your incoming class to be balanced to match the faculty interests. Also, you probably want to have applicants in a field to be given an extra look by the faculty in that field, as they might be better at assessments of the coursework, letters from colleagues in the field, etc. In summary: use broad categories here such as "number theory" or "topology", as being too specific can give an impression that you are not very flexible and have already settled on working some specific field. This can hurt your chances. Listing too many (small) fields is confusing to the administration personnel, who might have only people in the first two or three fields look at your application (in whatever order you are listing them), which may or may not be your strong suit.

(2) research experience. Predictions on what the person wants to do in the future are notoriously unreliable and tend to hold in the very broad sense (people who are interested in doing logic rarely end up getting a Ph.D. in geometric analysis, although stranger things have happened). What is useful is your previous research experience. My advice: after you mention very broad area where you want to work on, give very detailed description or your previous research in any part(s) of that field. This will show that your interest is not hollow and allow you to impress the graduate committee members reading your application.

About the length - try to keep it under 2 pages. Really. Make a webpage with all supporting material (like preprints of your papers, course projects, files with your research presentations, etc.) in case someone wants to look at it. Include the link at the end of your research statement. This will save time to those who don't care to read all this stuff and give ample opportunities for those who do.

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"...confusing to the administration personnel": I got a rejection letter from UC Berkeley saying that they regret they can't offer me a position in MSC 58, Global analysis (I may have written "Lie groups", which is MSC 22E, in my application) --- true story! –  Victor Protsak Jun 4 '10 at 4:40
    
Does the research statement have a value unless the candidate has CGPA in the upper most range? –  Anirbit Jun 4 '10 at 7:43
    
If someone's GPA is low, the exceptional undergraduate research experience is a good way to compensate for that. May not be enough, but it's worth the effort. –  Igor Pak Jun 4 '10 at 8:53
    
Actually I don't understand how GPA/marks from universities around the world is compared. Hence defining high and low is very hard. Colleges across the globe have different ways of marking and scoring. Somewhere GPA of 10/10 is common, at some places GPA of 9/10 is even hard to get, some have GPA in steps of 2, some have it in steps of 1, some have it out of 4. Some colleges don't even have a notion of GPA. They have raw marks quoted in the result. –  Anirbit Jun 4 '10 at 11:32
    
So if there is an application from some candidate whose marks is 65% in a college which quotes directly the marks obtained in the examination then is there any chance that this application will even be read? And about undergraduate research, I am reminded of a recent Gmail Buzz by Terence Tao in which he was talking of such matters and he seemed to de-prioritize the importance of undergraduate publications. –  Anirbit Jun 4 '10 at 11:34
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Igor's answer applies to US where you have 1-2 years to hang around a department before you choose a supervisor, and, maybe some other countries (Canada?). In majority of EU institutions, you will be given a Ph.D. supervisor straight when you come in. This makes your research statement crucial, as a professor will have to agree to take you on.

Igor's advice on writing two pages is applicable in EU too: nobody will read more than this. Make it a good narrative but avoid banalities from Wikipedia. Also remember that in EU, you are likely to attend an interview and if you cannot back your statement up by some background knowledge, it will look phoney.

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As has been noted, where you are applying might play a role; I will assume that you are applying to a US/Canada style school where the timeline is somewhat lengthened (4-6 years from admission to degree, at least) and where you might be expected to spend some time (a year or two) before "signing up" with a specific advisor.

I'll second most of Igor's comments: keep it relatively brief (no more than two pages or so), go into sufficient details in those areas you feel most strongly about to show that you actually know what you are talking about rather than just speaking generalities trying to sound impressive. I would also say that it would not be amiss to explicitly explain that your interests lie at intersections of fields (especially if you can make a clear and succint case for it showing you actually know what those interests are and why the many fields play a role, say one paragraph or two short ones at the most). It will not sound unfocused if you can make a clear case for why you want to have all those tools in your toolkit, or how you've used the varied toolkit in the past. As far as leaving stuff out, I would suggest picking around 3 major areas (covering as "wide" a territory as you can) in which you make your case, mentioning others. Of those three, again, pick one or two subareas in which you make your case, while mentioning others you might feel are important or you can say something interesting. But keep in mind the length.

Personally, I remember writing that I was interested in taking courses in several possibly disparate areas (logic, algebra, topology, analysis) which I had taken as an undergraduate, and making a case about the aspects I found common and appealing about them. I did get accepted on good terms everywhere I applied, so I guess it wasn't a bad strategy (back then, at any rate).

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I must remark on the last part of your comment. I have very, very brilliant and influential teachers with whom I have discussed admission to top programs (Michigan, Columbia, etc.), and who have stated in many instances that they would not have gotten into their respective alma maters if they had applied by today's standards. The field seems so saturated now with unusually strong applicants. I feel as though if I don't solve some major problem or graduate first in my class at my large university, I won't get a fellowship ANYWHERE... It's a bit depressing, actually. –  lambdafunctor Jun 5 '10 at 4:34
    
That's fair enough (you'll notice I said "back then"). I got into school in the early 90's; the situation had just recently gotten much harder (the school I got into had cut its admission in half from the previous year, and most schools were experiencing some sort of economic hardship and reduced fellowship offerings). The philosophy of graduate education and of admission was also different, according to many people I talked to, from what it had been 20 years before then. I don't know how it has changed there since then. I'm telling you the kinds of things I like to see in an admission packet. –  Arturo Magidin Jun 5 '10 at 18:22
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