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).