I have done some literature review on the target professors at each graduate school, and am considering whether or not to include a sentence such as the following, to hopefully demonstrate my genuine interest in their work:

"I am deeply interested in building upon Professor X's work to construct preconditioning methods for ridge regression with large parameters. In particular, I believe that methods A and B may be utilized in fashion C to tackle this unaddressed question in the literature... I am also fascinated by Professor Y's work..." and so on.

However,

• Since students should be open to many topics, could this be interpreted as being too narrow?
• Though I will try my best to avoid proposing a bad solution, is the risk worth it? I.e. is it often more likely that a student proposes a uneducated solution, than a good one?

For context: I am a prospective Statistics PhD student aiming for T20 programs in the US, with a math major at a T5 school in the US (good GPA, graduate math and statistics coursework, research experience, solid LOR). I have reached out to many of the professors at said schools, but none have replied.