How to judge a Ph.D. applicant that arrives "out of thin air"

I would tell them what you expect from a PhD student: prove theorems/run empirical tests/conduct human studies or all the above. If their profile is a clear mismatch, I wouldn’t bother.

I would next ask them to review two papers of their choice out of your recent publications. One page review each. The review should include at least one idea for future directions. It tells you whether they’re serious, what they want to work on, and whether they can reason about ideas in your field. It also guarantees that the problem is at least somewhat tractable (you’ve solved it after all).

If you’re happy with the write ups, schedule a video interview. You both are committing to 4+ years together, best to make sure you can discuss things together.

Good luck!

If the prospective student cannot realistically solve the problem - and it certainly seems so given that you + some colleagues couldn't solve it - then it does seem unprofessional to me. One could easily interpret your reply as a soft rejection, except it's dressed up to give the student some hope. Ambiguous signals are bad (just see how many literary plots involve one party in a romantic relationship giving ambiguous signals to the other).

I suggest instead using a problem which has already been solved, but the solution is not trivial. For example, you could take an exam question from a Masters-level course related to the student's field. The student should be able to solve it, although it will take effort. You can then assess how good the student is against the other students in the course. Alternatively, you could set some minor problem that will nonetheless need to be solved before a bigger one can be attacked. For example, "write a C++ program that numerically solves [this differential equation]" is not going to be easy, but if the student is able to do it, then you can maybe use that program on his thesis topic.

I'd avoid asking for something that takes more than several hours to do. The student is likely to be applying elsewhere as well, and it's not fair to expect him to sink a lot of time into one application.

There are an awful lot of very fine but disadvantaged students out there who just don't have any good network to connect them into good schools. I know some excellent folks who are now top professors and researchers only because somebody decided to respond to their "out of the blue" email.

Unfortunately, there are also vastly more mediocre or terrible students who will do nothing but waste your time. Moreover, these students are over-represented in perception because they email many more people than the good ones. Even just responding to their emails is usually a waste of time.

So how do I decide if a student who writes to me is likely to be worth responding to? Simple: did they write to me, or did they just write to an authority figure who might be able to help them? Some heuristics that I find helpful in telling the difference:

• Does the student talk about me at all, or just about themselves?
• Does the student show some familiarity with my work beyond just the title of a paper or two?
• Does the student explain how their interests may relate to mine?
• Does the student have some vaguely plausible ideas about what sort of work they would want to do together with me?

If a communication passes all of these tests, then it shows that the person has spent some real time to think about this communication and that there is some real reason that there might be a potential match. It's an honest signal of effort that has been invested, and that's a big deal, because it means they're writing to me and not just a list of ten thousand people with Ph.D.s.

Now, this doesn't actually tell you if the student is actually any good or not---for that, things like the tests that others suggest are not a bad thing to do. But this start gives some signals that seem to be fairly reliable in helping me distinguish between "application spammers" and potentially serious candidates to work with.