Should I list specific research questions and potential solutions for each professor in my SoP?

Question

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.

Answer 1

The trick with expressing interest is to mean it. If you say you are "fascinated" by a professor's work but then cite random papers and say vague things about them, that will do more harm than good. Even spending a few hours on the professor's website and skimming their papers will likely lead to only superficial comments that don't impress anyone (and this will require significant effort from you). But it's totally possible that in the course of your undergrad work, your work was related to Prof. X's, and now working with Prof. X would be a natural next step -- that's certainly worth saying.

So: I recommend being honest. If you are actually familiar with a professor's work because of your experience in this domain, you should certainly say so. If you're not familiar with the professor's work, but you understand the problem and some of the tools they use, that is perfectly fine to say. And if you have no experience at all in a given domain but are willing to learn, then it is better to admit this than to try to fake it.

Since students should be open to many topics, could this be interpreted as being too narrow?

It really depends on how it's written. Certainly there is a motif of "once you have a hammer, everything looks like a nail." At this point you probably only know how to use one or two hammers, so you should at least acknowledge that there are additional hammers that you should learn about in grad school.

Is it often more likely that a student proposes a uneducated solution, than a good one?

Much, much, much more likely. Your professors are very smart and have been working in this area full-time for years. It is extremely unlikely that an undergraduate, after a few hours' thought, can come up with a useful solution to their problem. So, best case, you'll give a plausible solution that they've already considered. Worst case, you'll say something ridiculous. Thus, I would not speak of "solutions" but rather of tools and approaches and interests.