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).
Best Answer
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.