A prospective graduate student is normally not hyperspecialized and inflexibly hyperfocussed on a certain well-developed research line (like a more mature mid-career researcher could be) just yet.

That is to say, even if such student worked on topic “X” (say, in the area of algebraic geometry) for their master’s thesis, they may well be flexible and willing to expand their horizon and start working on *any* interesting topic in algebraic geometry that falls within a certain wide range “A” (that possibly includes “X”).

So, I assume, *any* reasonably well-known and conveniently located school with “good” mathematicians that work on a quite wide range “A” of topics within algebraic geometry is in principle a good fit for an application of said student.

In this context, what does it mean to tailor a statement of purpose for a specific school? In particular, how can one do so convincingly (enough to have fair chances of being admitted) and professionally (that is, without spouting empty buzzwords or vague claims of interest)?