Vtgyjfy

I think that the best way to do so is to read one of their papers and ask them questions about it. If they’re able to confidently and enthusiastically discuss their work on a technical level then it’s likely theirs. Go attend their seminar and ask lots of questions!

As a math/theory person myself I can tell you that people often take great pride in their pet ideas and would often be very good at some things but not others. Don’t expect professors to have expert knowledge in every detail. In fact it’s usually the case that students are better than the advisor in some topic (this is the goal at least!).

As a final note, if you’re testing in an obvious/confrontational way many professors will rather not take you on. A successful PhD experience is a lot about trust and collaboration, so starting it out by trying to get the professor on the defensive will likely not succeed.

There is a story by Borges wonderfully illustrating the point. There, a prospective student comes to Paracelsus to study, but, before he would join him, he demands of him to prove whether it is true that he (Paracelsus) is able to convert matter from one form into another? This ends up with the candidate grabbing a rose from Paracelsus' table and throwing it into the fireplace.

The rose burns down to ashes and the candidate demands from Paracelsus to make the rose whole again. Paracelsus sadly shakes his head and the prospective student apologises for embarrassing him and leaves. After the candidate leaves, Paracelsus speaks a word and makes the rose whole again.

Moral of the story Even if I had the full set of abilities of a Feynman (who OP seems to value), I would not engage with a student that effectively demands from me to show them off. There are things that a student is entitled to know (such as supervision style, funding situation and similar), but effectively running an examination/interview for their prospective superviser is showing utter disrespect for the prospective supervisor's life experience, background and the fact that they simply can be assumed to have done quite a few things to become an academic teacher, whether strong in science or not (although, in an academic, without further evidence, a rule of thumb should be the "presumption of competence" - competent until found to be otherwise).

In short, the student is not the interviewer and shouldn't give themselves airs that they are. They should have the opportunity to judge their superviser via their publications and/or other visible activities. If that is lacking, they can have a conversation (not interview) with the prospective superviser to see whether they will get along.

Also, some supervisors have a number of students that are stronger than themselves (to name e.g. Sommerfeld or Feynman's supervisor, John Wheeler) - these are the best ones; they know how to get the maximum out from their students.

Bottom Line If the professor has a high reputation and you are not sure about their quality as supervisor, investigate whether their students end up successfully (e.g. publication record, visibility, academic or other career). Since you are considering joining their club, this is much more important than petty testing of the professor's specific skills.

I think that would be foolish to try. You aren't likely to find someone who is (a) good enough for your (in my view, ridiculous) expectations and (b) willing to work with you. Mathematics isn't a video game. It doesn't get advanced in quiz-like environments. And if the person is good enough, how will you be able to evaluate it anyway?

"Clever" isn't enough in mathematics.

What you want in an advisor is someone who (a) respects you, and (b) is willing to work with you. Someone with a good reputation could be a good advisor even if they are rather slow and methodical in their work. Someone who is "quick and clever" might be terrible as they leave you only scraps.

Surely you're joking Mr. Feynman!

(Sorry -again- for the final joke, but, your username ... Can't resist.)

Edit: after my answer was posted, the question was edited, changing the scenario from that of a student to the more general one of a “candidate” considering working for or with a professor. To clarify, my answer below addresses the original context of a student and may not be equally applicable to other more general situations.

My gut reaction on reading your question was to be taken aback by your perceived arrogance and naïveté and to want to criticize you for it, as the other (still excellent) answers are doing. However on further reflection I’ve come to believe that your question is actually more interesting than I realized, and I don’t think you’re necessarily either arrogant, foolish, or naive to be asking it.

In fact, what you’re asking is highlighting something interesting that doesn’t often get pointed out, which is that there is a big information asymmetry between professors and students. We professors are so used to this asymmetry that we take it for granted and don’t often stop to think about its consequences. We expect that students need to “respect” us and assume we are smart and competent because we have already proven our worth to lots of other people by publishing papers, applying for and securing a job in a prestigious university, and so on. But it’s interesting to be confronted with the reality that from the student’s point of view, none of this may matter: what if once in a while a student comes along who isn’t content with relying on other people’s opinions of us to form an impression? He (I’ll assume they’re a “he” because of OP’s user handle) wants to be convinced directly, by seeing it with his own eyes. In fact, Richard Feynman himself was very much just such a person who never gave much credit to “sociological” markers of prestige and would always seek to learn the truth, both about people and about scientific facts, “from first principles”. So I think your choice of user name is very apt, and, before I give my proposed answer to your dilemma, I’d like to acknowledge that I think your desire to become convinced through your own personal experience of a professor’s level of talent before you accord them the kind of respect most of them think they are automatically entitled to, is, in my humble opinion, very legitimate.

Now to address the question itself, which I’ll rephrase to avoid loaded words (that I think had a slightly triggering effect on some people reading this) like “test” and “interview”:

How can I become convinced, through my own direct experience, of a professor’s competence in mathematics?

The short answer is: you can’t; at least, not easily, and certainly not in a single conversation. This goes back to the information asymmetry I pointed out above. Imagine me taking Japanese lessons for two weeks and then meeting a new Japanese teacher and trying to decide if they do in fact speak correct Japanese as well as they claim to. There’s no chance for me to be able to say anything meaningful about the person’s Japanese, is there? I just don’t have the knowledge that’s needed to make that sort of determination. It’s similar with math - a math professor has had so many more years of training and experience than you that you are simply not in a position to decide directly if they are “clever” at math (even if we adopt the premise that you get to “test” them in a way that doesn’t immediately put their ego into defensive mode, leading them to kick you straight out the door). They will likely find any question you can ask them either very easy or very naive, or both.

Now, if you interact with someone over the course of a few months or a year and watch how they handle a variety of questions and problems, including things that are research-level problems or from areas of math outside their expertise, there is a good chance you’ll be able to develop a feel for their level of “cleverness”, or at least the very specific type of cleverness that involves “thinking on your feet” in a spontaneous setting in the presence of other people (something which some really good mathematicians nonetheless aren’t very good at doing). But as I said, that will take a much longer time than just a single meeting.

To summarize this long answer, I think it’s good that you’re a Feynman-type person who insists on reaching their own conclusions through direct experience. It’s an unusual trait, and one that could serve you well in your scientific journey. But there are situations when you’ll need to have a bit of humility to accept that there are facts about the world that you lack the knowledge to intuit yourself, and you’ll have to rely on those indirect sociological cues that you normally prefer not to rely on.

Anyway, good luck!

I agree with a lot of the answers so far, but I want to point out one other relevant point:

In academia, unlike in industry, interviews are generally not used as a way to evaluate a person's technical skills or abilities. Instead, such evaluations are usually based on:

1. The person's documented achievements in the field; and




2. Recommendations from other people whose expertise in the field is recognized, and who are familiar with the person's work.

This applies at pretty much all levels of academia. When departments hire professors, they do not ask them "clever" interview questions to evaluate their technical expertise; rather, they look at their papers and talk to other experts who know their work. The same goes when professors hire postdocs, or when professors hire grad students (there may be a few exceptions to this last one). The interview is meant more to evaluate the person's "soft skills", administrative and organizational abilities, and to discuss the person's achievements and goals in depth.

I won't really go into the reasons for this practice, or attempt to justify it, except to say that the qualifications for success in an academic position are typically much broader, and much less specific, than for, say, a programming position, where they really just want to know "Can you code in language X?"

So this explains why people are finding your question somewhat outlandish: professors don't ever get asked "clever" technical questions in any kind of an interview, because nobody finds that to be a useful method of evaluation. Thus the notion of having a student ask such questions of a prospective advisor sounds particularly absurd.

Now, I think it's perfectly reasonable for you to want to have some idea of a professor's qualifications in a certain area. If you want to learn about a particular area of math, then you'll want to try to identify an advisor who can help you in that regard (though, as a graduate student, you'll also be expected to develop the ability to learn independently). But you can go about it in the same ways listed above. Look for achievements: Does the professor write papers that use the sorts of techniques that you want to learn? Do those papers seem to be influential? And look for recommendations: talk to the professor's current and former students, and look at their records. Did they learn the sort of things that you want to learn?

Of course, you can also ask the professor directly: "I am interested in learning about theoretical area X. Is this something that your students typically learn? What does that process look like?"

I think your prior is wrong. Lots of people want to become professors, but only a few succeed. Therefore instead of assuming that the professor is weak and should prove his/her strength, you should do it the other way: assume the professor is strong until proven otherwise.

If you want to test something else, like whether your personalities are compatible, whether the professor is not so busy that he'll never have time for you, etc, there're ways to do that. But if you want to test if the professor knows math ... try reading your OP again, suitably edited.

I'm an undergraduate who needs to find a PhD student mentor in an undergraduate research project.

Some PhD students might have many papers and a high reputation, but aren't necessarily good at theory/maths. In a face to face or video interview, what will be the best strategy to interview/test the PhD student's maths/theory ability? It's of course usually the other way around in that the PhD student wants to test a undergraduate's ability. But a strong undergraduate should also look for a strong PhD student through clever interviews.

No surprises if you're offended.

Pretty much any professor is competent enough to advise a PhD. If your PhD advisor isn't up to your standards in clever, you can fix that by talking to other people or getting co-advised.

It's not that you shouldn't be testing the person who you're going to pin your professional career on, it's that you shouldn't bother to test them for things that don't matter.

As a student you shouldn't worry about competent. You should worry about crazy. That's what the interview is for. For you as a student to figure out if this person is someone you can live with for 5 years. If your PhD advisor is nuts you're probably just screwed. No matter how clever you are, or think you are, you can't fix crazy.