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What are the 'right questions' to ask professors at a visit day for prospective graduate students?
I'm in mathematics, though I don't know if the answers to this will be field-dependent. There have been previous good questions that answer what to ask graduate students here and only somewhat relatedly here (about what to ask students of potential advisors). However, I don't know what good questions are to ask professors themselves. The big question is, of course, "will I be happy at this school?", but it's not clear to me what the 'right questions' are to help me get a feel for this.
Essentially, what should I know about the program/school that are suitable questions to ask a professor during a meeting at these visit days?

Many many years ago, when I was a prospective graduate student, talking to professors during school visits was terribly awkward (at least partly because I had no idea what these conversations were supposed to be like - if only AC.SE had existed back then!) Of the four schools I ended up visiting, at two it seemed pretty clear that most/several of the professors were participating in the 'visit day' quite grudgingly, and at the other two, they seemed close to universally enthusiastic about meeting with us (anecdata: the latter two were small departments, and the former two quite large)...

...This was a (small) part of why I decided to go to a smaller (if not as highly-ranked) department. I can't speak to whether I would have been happy at the other schools, but I have had an entirely positively experience at grad school, in no small part due to the faculty being so interested in the well-being of us students. [Of course, I agree with Pete L. Clark that awkwardness in visit day conversations does not imply lack of concern about well-being of students]

"Is the department making you do visit day duty against your will? Blink once for yes, twice for no."

Good question.  I am a tenured (associate) math professor at UGA.  To be (too?) honest, I sometimes find graduate visitation days to be slightly awkward.  With most of the students, they and I want to have something to talk about; for many of the students, we will have something to talk about at some future point; for a few of the students, we will become so close that it will be hard to remember these first awkward conversations.
I just remembered a visitation day where I almost sullenly plopped myself down at a barroom table next to a certain prospective student and a bunch of current students.  They were more trying to have a conversation than actually having a conversation.  At one point the topic turned to some elementary topology problem the prospective student had in mind for some reason.  One of the current students was studying topology and she got interested and started to try to work it out.  Alas I was still bored, and after a few minutes I quietly uttered a few words.  The current student paused for a second, then her eyes lit up, and very soon after I got my chance to politely excuse myself and get another drink.  That prospective student is now my PhD student, but the bond was not forged on that day.
But still, we should try.

1) I hear you study X.  I've studied only a little bit of it so far, but I remember learning about Y.  Can you tell me a little about your work?

"Can you tell me about your work?" is a very natural question that sadly can often be a showstopper.  The honest answer for many professors when talking to perfectly capable students who may later turn out to be their students is often "Not very meaningfully, no." Above I tried to frame the question in a little better way.  How is it better?  Well for one thing, asking a professor in a department you're visiting what they do can come off as lazy: almost everyone has a wealth of information available on their webpage now.  If the student really cared, wouldn't they look a little bit in advance?  So the framing of the question shows that the student has looked a little bit in advance.  It also clues the professor in to what might be an appropriate level at which to pitch the explanation.  If X is number theory (as it is for me), then if Y is "Wilson's Theorem" I'm going to say something very different than if Y is "the Chebotarev Density Theorem".
Still it's probably best not to say too much.  I remember asking one professor this when I was a prospective student and he said, "Yes, I study number theory, as do several other faculty members here.  I guess what distinguishes me is that I've also become very interested in representation theory, and my work uses that to a larger extent than the others.  Does that make sense to you?"  The person who said this to me was Dick Gross, and yes it did make sense, and as a two sentence orientation to who he was and what he did it functioned brilliantly then and it still does, but I see now what a lot of confidence and experience he must have had to know to say so little.  And he asked me if I understood, and if I hadn't he probably would have said something else: he seemed like a nice guy...and it turns out that he is.  (I don't remember what I said after that.  I think it would have been like me at the time to try to show some knowledge by asking a question -- that's a young math student type of thing to do.  So I'll imagine that I asked "What kind of representation theory?" because I would have known that there were finite groups, Lie groups and so forth.  And Gross would have replied "Algebraic groups".  At that point I would have had to admit that I was totally lost...and my being lost would be of no lasting significance: suffice it to say that I have since learned about that topic!)

2) Do you do any computer work?

I think that's an increasingly important question.  Nowadays a math student has to locate himself on the spectrum of more computation / less computation.  If you're thinking about working with a faculty member, you definitely want to know this.

3) Are you interested in real-world applications of your work?  Have you ever done any consulting or other outside work?

Again, a key question.  You should already be starting to think about these sorts of questions for yourself.  I, for instance, have absolutely zero contacts in industry, so I would be a terrible advisor for a student who was inclined in that direction, even if we had common mathematical interests.

4) How long have you been here?

That's a nice question for one human being to ask another.  The professor will probably respond with some personal information about the town or their housing situation or their family or...something.

5) Where did you get your PhD?

You would think that would be a kind of backward-looking question to ask a professor, but actually academics ask each other that kind of thing all the time.  Sad to say we really do try to have little portions of the CVs of hundreds of people memorized, even though we can look much of it up instantly if needed.  Also this question subtly asks the professor to put himself in your shoes.  It may also lead to some kind of insightful contrasting between the professor's PhD experiences / program and the current PhD program.  On the other hand many faculty did their PhD at better places than the ones in which they are currently employed, which can be a little embarrassing.  But there are various ways to spin this; maybe a temporal comparison is more interesting than a programmatic comparison.  If a faculty member is old enough they can tell you that their graduate stipend was 75 cents a day plus all the potatoes they could carry, or something fun like that.

6) Do you have any PhD students?  [Then ask plenty of followup questions.]

That's getting serious.  You want to find out whether the professor is even on the table for being a potential advisor.  And if a professor names his students, you can then go and seek out those students and talk to them about their advising experience...that can be very enlightening.

7) Are you happy here?

That's a bold question, and you might try to come at it a little obliquely, but again it's a very standard question for academics to ask each other.  Currently academia has an itinerant feel to it: we are all looking elsewhere around the terrain and asking each other, "Hey, long time no see, could you show me a blade of grass over there?  Hmm, that's pretty green, right?  I mean, do you think?  How does it compare to this blade, would you say?  Greener?  In what ways?"
Sometimes when you ask this question you may hear that the professor has already made or is currently making plans to leave!

8) How would you compare this program to other programs Y and Z?

Again, we're good at this, by and large.  Even if we inflate our own program (it seems to be called for to do this at least a little), we are likely to be quite honest and rather knowledgeable about other programs.  If you go to five different places and ask around, you're likely to get some good consensus data.

9) What advice would you give to a student who wanted to be successful in your program?  Are there any specific pitfalls to be avoided?

That's an obviously good and relevant question.
Note that you can't ask "Will I be happy here?" for a simple reason: we don't know.  If you're lucky, we know here and we know us.  We don't know you.  I think that you can in some ways get better information about the answer to this question by talking to students than to faculty, but any way you slice it, it's something you're going to have to largely figure out for your own.  No one at the visitation day is going to know you like you do, after all.
Good luck.

This is very helpful, thank you! I do feel a bit wary about #6 and #8 as you mentioned - it feels like being too forward too soon. The other ones are certainly more than enough to hold a good conversation, though.

@Mike: I just want to check on this: are you saying that you think "Do you have any PhD students?" is too forward?  I really think you shouldn't.  You're asking something that is (i) totally public information, and at the same time (ii) very much expected that you will be asking that.  I wouldn't feel shy about this at all.

@PeteL.Clark Yes, that's what I was saying - my impression was more that visit days are about determining how one would fit at a university rather than with a specific advisor, which is what this question seems to be asking about. But I freely admit that I am sometimes an overly cautious person.

@Mike: That's a question that any student can ask to any faculty member at any time, and it really need not mean that they are fishing for a future advisor.  Why do you talk to faculty at all?  You want to know how faculty view the graduate program and graduate students, right?  You'll learn the answer to this question best by hearing faculty talk about the students they advise.  (If a faculty member has never advised any students, then they're not the ones whose interaction with the students is significant, at least to a large degree.)

@PeteL.Clark "...then you certainly want to know that when you consider everything else they say." That's a very good point that I hadn't considered. Thanks for your input and your time!

I sympathize with the OP. The biggest issue I have is that I comprehensively stalk cough research any potential advisor that I want to work with, so I know the answer to the everything but 7-9. Guess it's never bad to ask..

Some questions that I think are all too relevant to mathematics PhDs are
What percentage of entrants get the PhD?
How many years funding are there?
What percentage of students finish before funding runs out?
How do those without funding survive?
Does the university allow ABD status and so no fees?
Where do graduates of the program go?
At what stage are you confirmed into the program and what do you have to get confirmed? eg quals or orals?
Are supervisors required to meet with students regularly? how much time do most supervisors give to their students?
How many students are in the program? what are their backgrounds?

Although understandably so, all these questions are unrelated to the realities, despite sounding like they'd be exactly the thing... I'm not saying this to be mean, but to avoid people getting the impression that this answer hits the mark.

I have to agree with Paul Garrett. You should try your best to get the answers to most of these questions, but I wouldn't ask a faculty member any of them during the interview.

These are good questions to ask the students in the department -- but not the faculty.

these were the important questions that were so obvious to me when I finished the PhD and so completely non-obvious to me when I started. I am now a full professor and the PhD director for my program. I would not regard any of these questions as inappropriate if a student asked them of me. You don't say why you think they are unrelated to the realities.

Belatedly: first, faculty often do not know the answers to any of these questions,  (though may misguidedly believe they do!?!?!) and would not be authorized to commit to anything. And what does "confirmed into the program" mean? And.. statistics about advisors meeting advisees, or, worse, requirements, are not-to-the-point. And info about the backgrounds of students already in the program would be unavailable to nearly all faculty, and would be essentially illegal to divulge, from those who did have the info.

(That is, sure, it would be interesting to know the answers to these and similar questions, but faculty mostly won't know the answers, etc.)


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strategies to secure more independence in a postdoctoral position?
I am soon to finish a long PhD, and am thinking about postdoctoral applications.
As a PhD student I did a lot of quality work and have some publications in respected journals/conferences. However, I did not develop my own research ideas or objectives as a PhD student. This was just the state of affairs and there wasn't much I could do about it. Also, while my work so far is primarily applied and methodological, I am interested in theoretical work. I do not want to continue doing the kind of work I have done so far.
I have a couple questions under the theme of the title:

Do I need to sell my PhD work as my own idea in a postdoc application, or is it reasonable to be honest?
Will it arouse suspicion in an application to propose research in an area which is not closely related to my PhD research?
If I want to be able to develop my own ideas as a postdoctoral researcher, should I seek to clarify that goal, seek positions which advertise that option explicitly, or conceal that goal and instead simply do so once I have a position?

More general question:

From my observation and reading, there's an art to developing a research objective/question. Ideally, I would like to have some kind of guidance in doing so. In my experience that is unrealistic. How can I seek meaningful mentorship, but somehow frame it in a manner that allows me to move forward and to maintain financial and intellectual freedom from the mentor?

In my experience for students from mathematics and physics the answer to (2) is often "no". However, coming from an applied area wishing to do more theoretical work, the standards are unclear to me.

Welcome to Academia.SE! Your questions are interesting, but could you please use a thread per each single question? (StackExchange system does not work well for putting multiple questions, even if related, in one thread.)

@PiotrMigdal I would say these questions are related enough that they should work well in a single thread.

I work in a completely different field, so I am not going to answer your questions lest I say something completely wrong. That being said, in my experience, being entirely honest about your goals and expectations is always the way to go in academia. Academia is a small village - when you deceive your current mentor, the next one (or the next faculty search committee) will know about it.

Would you consider being a little (or a lot) more specific about your field? It probably doesn't matter much, but might help to clarify the question.

I am in a computational biology subfield -- one that's small enough I'd rather not name it specifically. My work has been in developing methodology and performing data analysis.

@xLeitix I would certainly prefer an honest approach, and to work with people with whom I could be honest. However it has become my impression that it's expected that you will not be honest, and in fact that honesty is considered naive and may hold you back. People are sometimes described as "too nice". Perhaps that's just my field, though, I don't know.

@user11948: I make a mental notice of people who say other people are "too nice" and then make sure I stay far away from them. IMHO this tells more about the one who speaks than about the one who is spoken about. Also, if I get the impression that someone expects dishonesty (in job interview questions - not about students during an exam) I tend to conclude that they may be dishonest themselves - and that's another one with whom I don't want to collaborate.

@user11948 I guess there are almost certainly people that consider me "too nice" or naive as well. However, I have yet to find a job (TT or postdoc) where they would rather hire somebody with the reputation of playing political games than somebody with the reputation of being a straight-forward researcher. Is it possible that you are just rationalizing for yourself why it would be ok to deceive your future mentors?

@xLeitix Well, something like that might be true. However, I think deception is a strong word. I don't really think I have anything to gain personally by 'deceiving' anyone. I just want to understand the breadth of expectations. Just for instance, before coming to academia I never would have dreamed that author order was such an important thing. But it is. It's this kind of thing I'd like to figure out.

@xLeitix I should say, I come from a "lab" field where professors rely on graduate and postdoc labor for advancing their own career. Early in graduate school I was told by a senior professor that he thought one might have to "break the skulls" of a few graduate students/postdocs in order to get tenure. This holds true in retrospect. My observation is that in other fields, like say mathematics, this is somewhat less often the case.

@xLeitix It's in this context that I voice my concerns over "financial and intellectual freedom". And so I do feel my questions are substantive.

First of all, let me say that your choice of words really rings an alarm bell with me. The general impression is that you are very ready to be inhonest, and (in agreement with that) your default position seems to mistrust your future employer and you seem to expect suspicion as opposed to taking in the situation openly.
Now if someone does not trust me, a natural question for me is: why should I trust them? I would not want to work with someone with this attitude towards work. And moreover, I wouldn't want to have someone messing up the working atmosphere in my group.
(But don't worry: I'm not in the position to hire anyone.)

Do I need to sell my PhD work as my own idea in a postdoc application, or is it reasonable to be honest?

I'd say it is even necessary to be honest, and it is most probably futile to try to get away with anything else. Academia is a small world, and phone, skype and email reach very far nowadays. Particularly if you say that your field is so small that you do not care to name it here.

Will it arouse suspicion in an application to propose research in an area which is not closely related to my PhD research?

No. But you should have a positive reason to apply there.

If I want to be able to develop my own ideas as a postdoctoral researcher, should I seek to clarify that goal, seek positions which advertise that option explicitly

I'd say that you are expected to develop your own ideas in a postdoc position.
So: Yes, why not. I had interviews where we discussed openly how much own ideas would be possible, welcome, and what the bottomline of things-that-need-to-be-done-no-matter-what would be.

From my observation and reading, there's an art to developing a research objective/question. Ideally, I would like to have some kind of guidance in doing so. In my experience that is unrealistic. How can I seek meaningful mentorship, but somehow frame it in a manner that allows me to move forward and to maintain financial and intellectual freedom from the mentor?

I can ensure you that there are good mentors and leaders, including also mentors and leaders who are even good at teaching leadership in research. But such learning can only work if you trust your mentor. That in turn makes financial and intellectual dependence a non-issue. It may not be easy to find a good mentor. But on the other hand, you could also learn from someone who is not your direct supervisor. That way, you'd have the financial and intellectual freedom.

But: if you feel you need mentoring how to develop research questions, how can you feel ready to apply for a postdoc position?

I did not develop my own research ideas or objectives as a PhD student. This was just the state of affairs and there wasn't much I could do about it.

How come? How could your supervisor prevent you from thinking your own thoughts and from having your own opinion and judging of what needs to be done and how? As a research professional, crititcal and independent thinking is one of your core tasks.
Remember: you were a professional already when you started the PhD. If that wasn't necessary, it would be appropriate for an apprentice to apply for a PhD position.
And if you had gone to work in industry instead of in academia, you'd aslo have been profesionally responsible for everything you do.

Hi. Thanks for your answer. I don't feel my question implied "dishonesty". People "spin" their message. I am asking how best to do so. Further, in my field it's very common for people not to have worked on their own questions in their PhD. Writing a PhD is literally a matter of stapling together your papers. Further, I know many postdocs who are just like graduate students: they are unable to work on their own projects. Hence my question.

"How could your supervisor prevent you from thinking your own thoughts and from having your own opinion and judging of what needs to be done and how?" That's silly. I was paid by my advisor and expected to work on his research problems. That's my job. I can assure you I am "professionally responsible" for my work. I didn't say I didn't do my own work. I said I did not choose the problem.

"But: if you feel you need mentoring how to develop research questions, how can you feel ready to apply for a postdoc position?" Because I am graduating. I will work on my own problems. However, it seems natural to seek some kind of guidance, especially since I want to enter a different field.

I'm sorry you didn't like the tone of my question. However, your rejecting that tone does not remove me from the context in which it was written.

Time for some tough love.  What your future employers want is a steady stream of scholarship, increasing in quality, quantity, and self-sufficiency. This includes both those hiring you for a postdoc and those hiring you (hopefully) for a permanent position afterward.
Normally to switch from subject A to subject B requires time and produces a gap in the stream.  This is very bad.  Even if there is no gap, switching topics renders much of your hard work publishing papers less relevant to future employers.  All other things being equal, I would prefer to hire a specialist in subject B rather than someone who straddles both B and A.
If you're Terry Tao, you can work your way through the entire MSC 2010; that's just being brilliant and prolific.  However if you're a mere mortal it's a bad idea to switch specialties until at least after tenure.
PS. If you are less than forthright about how experienced you are at working independently, this cannot possibly lead to a result in your favor.

Do I need to sell my PhD work as my own idea in a postdoc application, or is it reasonable to be honest?

You need to honestly show how your work is your own and you need to show that the research question you worked on is important. The people hiring you want to see that you do groundbreaking work on important problems, they aren't looking for a story about how an original research problem came to you in a dream. You need to talk about what you did. You should devote very little space to what other people did, if any.

Will it arouse suspicion in an application to propose research in an area which is not closely related to my PhD research?

Not if you justify your interest in the proposed research. It would be good to show some sort of conceptual connection between the proposed research and the previous research, but probably not essential if the proposed research is something "hot" in your field.

If I want to be able to develop my own ideas as a postdoctoral researcher, should I seek to clarify that goal, seek positions which advertise that option explicitly, or conceal that goal and instead simply do so once I have a position?

Developing original research ideas (under constraints) is usually the point of a postdoc. It's not really something you need to clarify, and not something that needs to be in the ad. Do make sure that the ad is compatible with your intended research direction, obviously.

From my observation and reading, there's an art to developing a research objective/question. Ideally, I would like to have some kind of guidance in doing so. In my experience that is unrealistic. How can I seek meaningful mentorship, but somehow frame it in a manner that allows me to move forward and to maintain financial and intellectual freedom from the mentor?

The postdoc isn't the PI. At a postdoc position there will usually be a few senior people who can be your mentors. So you'll get mentorship.
Meaningful mentorship isn't exactly something you can ask for (or that can be given on-demand). It's like friendship: it doesn't really make much sense to "ask for meaningful friendship," it's something that develops with people who are the right fit for you.
Intellectual freedom is sort of implied in the position, as mentioned before. The postdoc's job is to do original research on a particular topic. Obviously there are constraints to what research you can do, but that's true at all levels: everyone has constraints on the research they do.
As for financial freedom... I'm not entirely sure what one needs to do to get fired from a postdoc, but I think that having original ideas isn't that thing.

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Are there widely accepted rules related to union officers being on faculty senate?
The faculty senate at my school recently took a position on a professional issue that was highly confrontational toward the administration. After union negotiations resulted in a pay raise, the senate reversed itself on this issue. It was later revealed that the original position had been advocated by the union as a way of applying pressure in contract negotiations. Some but not all members of the faculty senate knew about the union's advocacy of the action. There was no direct or logical connection between the professional matter (the hiring of a vice chancellor) and the contract negotiations. A large number of union officers are also on the senate.
Do other schools have rules or standards of ethical conduct that cover this type of conflict of interest? Is it considered normal for there to be so much overlap between the union leadership and the senate, or for the senate to act so closely in concert with the union as part of labor negotiations?

This is going to depend on institution type and location, and the culture surrounding shared governance. As an example, where I am, faculty are not union.

@BenNorris: This is a community college in California. We're governed by California's Title 5, which states that faculty senate is supposed to be concerned with a certain list of "10+1" academic and professional matters.

Many faculty senates agree to be governed by Robert's Rules of Order. Buried in it is this clause:

No member should vote on a question in which he has a direct personal
or pecuniary interest not common to other members of the organization.
For example, if a motion proposes that the organization enter into a
contract with a commercial firm of which a member of the organization
is an officer and from which contract he would derive personal
pecuniary profit, the members should abstain from voting on the
motion.  However, no member can be compelled to refrain from voting in
such circumstances.
Citation: http://www.cityethics.org/content/roberts-rules-has-conflict-interest-rule-local-governments-no-conflict-provisions

Note that it says that members cannot be compelled to abstain and so there was no violation per se. So in your case, I think the best thing to do would either to work within the Senate to all for an inquiry possibly leading to an impeachment or vote of no-confidence in the leadership.
If you want to avoid direct confrontation, you could leak this to the administration which would be happy to do the inquiry for you, but this would be seen as highly uncollegial and quickly make you unpopular.
How about asking the Chronicle of Higher Ed to look into this as a third alternative?
Edits: Clarifications and obfuscations.

In short: no. However, there is one rule that most higher education I've encountered seems to stick to: they really like constitutions and written rules.
A quick google of my own university found a faculty senate constitution of over 30 pages with a great degree of detail and rules, and absolutely no mention of conflict of interest.
From further review, I might propose an idea of why: interest is assumed. Rather than being an organization who seeks impartiality, there is actually a deeply vested interest assumed that all senators - and the senate as a whole - try to forward the goals of the university and the interests of the faculty itself. Using my own reference constitution, early on this line states the purpose of the Senate: "The Senate shall be the representative body of the faculty." In other words, they are out for what is best for the faculty.
As such, in the situation you briefly outline, there isn't a conflict of interest, but it sounds like a case instead of "vested interest" - the faculty, and thus the senate, want pay raises and oppose administration that seeks to oppose or refuse it, and so they do what they can to help achieve this mission. In short, this isn't unethical - it's the definition of politics and the reason why the senate exists.
Much of this is determined, at least in the US, by state law and mandate for public universities. In Wisconsin: The faculty derives its authority from 36.09(4) Wisconsin Statutes, which reads as follows:

FACULTY. The faculty of each institution, subject to the
  responsibilities and powers of the board, the president, and the
  chancellor of such institution, shall be vested with responsibility
  for the immediate governance of such institution and shall actively
  participate in institutional policy development. As such, the faculty
  shall have the primary responsibility for academic and educational
  activities and faculty personnel matters. The faculty of each
  institution shall have the right to determine their (sic) own faculty
  organizational structure and to select representatives to participate
  in institutional governance.

As with all good representative systems, it is assumed that there are checks and balances. There are usually student government wings, faculty senates, administrative positions, committees, etc - all designed to, hopefully, ensure an outcome that is the best for the whole community.
But in the case of a faculty senate being in favor of pay raises and also being pro-union? Yes, yes I would imagine they are. Whether or not they get a raise is usually not solely determined by a faculty senate, though, or that would be a pretty naive system.

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Is doing two PhDs a good path?
I am studying a combined bachelor of engineering (electrical) and bachelor of mathematics; I just started this year and will graduate in 2018. The reason why I am doing double degrees and not a single degree is because I love both electrical engineering and mathematics and I could not ignore any of them. So with this in mind, I am thinking of doing two PhDs when I graduate (one in electrical engineering and one in mathematics). Is this a good path or I should concentrate on only one of them?

You just started your double major. I think it's too early to plan PhD right now. Concentrate on your bachelor of engineering and bachelor of mathematics. Three years later, you'll have much better idea if you really want to do two PhDs. Then ask this question again.

I wouldn't worry about it. A surprise twist will land you in chemistry or music or something unrelated anyways.

A good path to what?

You're asking for advice on a course of action without stating what your goal is. If I asked you "should I take the bus?" I hope you would point out that you don't know where I want to go. Do you want to have a career in academia? Do you want to have a career in industry? Do you want to have a lot of certificates on your wall? What's your goal?

I suspect that most people who ask about the possibility of doing multiple PhDs are missing something fundamental in what a PhD is and what it's for.  This is an understandable misperception because the general populace knows little about what PhD programs are all about, to the extent that even most people who enroll in a PhD program think they know what they're getting into and then find out that what they had in mind is an approximation to the truth (sometimes a good approximation and sometimes not).
In the various fictive media (novels, comic books, television, film...) having multiple PhDs -- often in confluence with a very youthful persona or explicitly pointed out that they were attained at an early age -- is a standard trope for a certain type of characterization: depending upon the genre it can signal super-genius types, intellectuals, or nerdy/socially awkward types.  For instance, Professor Charles Xavier graduated from Harvard at the age of 16, and he holds PhDs in Genetics, Biophysics, Psychology, and Anthropology as well as an MD in Psychiatry.  This is obviously meant to be a real-world grounding for his vast mental powers.  On the TV show House, the youthful medical student Martha Masters has previously attained PhDs in applied mathematics and art history, and the characterization is less positive: they are not trying to suggest that she's a genius per se -- House is the genius! -- but rather that she has an extremely overachieving personality type.
Of course, Charles Xavier and Martha Masters are fictional characters.  Moreover I imagine they were written by people who are not so familiar with PhD programs.  In real life, having PhDs in Genetics, Biophysics, Psychology and Anthropology is not the hallmark of a super-genius: rather it would mean that the person is pathologically obsessed with graduate study.  PhD's are not merit badges -- more of them is not better.  Rather a PhD is the necessary and grueling training for a specific type of professional career.
If you are interested in a subject then you read and learn about it.  You get a PhD in a subject because you want that subject to be an essential part of your career (which of course ought to imply that you are very interested in the subject).  Moreover, although the PhD provides specific technical training in a certain subject, it does not provide exhaustive training: in most cases people spend the rest of their careers continuing to train in these areas.  Rather the point of a PhD is to give you enough specific training so that you can henceforth direct your own training and learning.  Otherwise put, a lot of what you do in a PhD is learning how to learn.  But learning how to learn is actually a rather robust and subject-independent skill.  If you learn how to learn genetics, then if later on your interests turn to psychology or biophysics you will be in a much different and better position to train yourself in these areas.  In some ways, doing a PhD in one of these subjects and then turning around and doing another is like becoming an internet millionaire who wakes up one day and decides that she wants to sell electronics...so she shuts down her website, gives all her money away, and starts the business out of her parents' basement.  (Or maybe it is like what happens in some of my dreams: I am doing high school all over again, and somehow it is not going as well as it did the first time around.)  Why are you starting over from scratch?!?
It is true that "academic transfer" makes better sense between some fields than others.  If Martha Masters got a PhD in art history and then decided that she wanted to do applied mathematics instead then she would not have been able to "segue" from one field to the other: she would indeed have had to go back to school.  But that makes her story a bit sad: getting a PhD is very difficult and very time consuming, but it is not really a feat of strength, and getting two PhDs in two unrelated areas does not show how much of a super genius you are; it shows that you really changed your mind considerably about what you wanted to do with your life and maybe wasted a lot of your time.  No matter how smart you are, I don't see how you can get a PhD in art history and then one in applied mathematics without spending at least eight years.  You can't just skip to the end and pass the thesis defense by virtue of your preternatural brilliance: there is coursework, residence requirements, and various other mandatory things which necessarily take time.  Her character is comic-booky: it is not really plausible that she could have these two degrees and have gone on to be a medical student while still being in her 20's, no matter how brilliant.  [Well, "no matter how brilliant" is a little too strong: if she had graduated college at age 16, then the math works out okay...] Moreover, for someone who is that brilliant it would really be sad that she can't seem to figure out anything else to do besides infinite schooling.
Electrical engineering and mathematics are rather closely allied fields.  For sure do a PhD in one or the other.  While doing a PhD in either one you can choose the amount of involvement you have in the other field, i.e., you can do a very mathematical electrical engineering PhD or a math PhD on a mathematical topic with important applications in electrical networks.  Then after you get your PhD you can continue to learn and train in one or both fields as you see fit.  It is entirely plausible that you could land an academic job in one department while having gotten your PhD in the other department.  This is an ambitious goal, but any academic job is an ambitious goal.  Getting the second PhD is unlikely to be directly helpful once you have it, let alone worth the sacrifice of 4-6 years of your life!
Added: By the way, when you mention "mathematics" and "electrical engineering": I think of Raoul Bott, a brilliant and beloved professor whose long career at Harvard ended while I was studying there.  I think that his story will be inspirational for many people with this confluence of interests.  Check out his bio: he balanced his early interest in these two fields nicely...and only needed one PhD.

Very well put, I totally agree. There is one exception, though, the PhDs that are awarded honoris causa. Having one PhD, the second should be pursued this way, IMHO. It's even arguable that the first one should be pursued this way, basically doing real research and contributions (those are real credentials) and not coursework that implies that you can memorize some lessons but does not necessarily imply that you can do some research contributions. The main (only) purpose of the coursework should be teaching a person how to do these contributions and that's what should matter to award a PhD.

Indeed, once you have a PhD, there are much easier ways of getting the benefits of another PhD than actually doing it.

To emphasize how mathematical Electrical Engineering is, the Dean (or Director?) of the college of Mathematics at my school has his undergrad in Electrical Engineering. As you'll learn if you don't already, there is a wide range in every field between the highly theoretical and the highly practical (think EE vs EET) and, of course, every field covers a wide range of material (EE covers everything from the borders of CS to Mechtronics, but this is hardly exhaustive). As Pete is saying, you can make your degree more or less Math heavy by taking those courses.

In some places of the world, there is no such thing as "grad school" and the only thing you have to do to get a PhD is hand in a thesis. So, while you probably won't (want to) do two PhDs by going to grad school, it might make sense to write up a thesis (i.e. book) about a new area you moved into and thus get a second PhD -- incidentally.

There are universities that award "PhDs by publication" in at least some subjects. I don't know if that happens in the US, but perhaps we can cheer ourselves up by imagining that Martha, while following a typical PhD program in one field, published valuable peer-reviewed papers in the other field and then (perhaps due to her juvenile over-achievement instincts) opted to pursue PhD accreditation for them which cost her relatively little time. When it surely wasn't necessary: once she's in medical school what range of job choice is she trying to keep open? ;-)

... of course this doesn't provide a template to follow in real life, because your PhD program in point of fact will not leave you time to throw out novel research in another field in your lunch breaks, let alone getting it published.

@Steve Jessop: Who says you have to do the second phd at the same time as the first? I actually have a friend who got her phd and went on to a post doc in a related but different field where she first-authored some well recieved papers and then decided that she'd like to stick to that field and wrote up a "cumulative" phd thesis out of those papers. She ended up with 2 phds at age 28. I do admit that it's a very rare course of action.

It should be noted that a lot of people don't care what a PhD is "for" when they take it, just like many people don't care what food is "for" when they eat it: and these are often the most well fed!

This answer seems very American to me. In Europe, I think, even in Academia, multiple PhDs are much more valued.

To this excellent answer I just wanted to add that in Academia, the exact opposite of the Hollywood trope often applies: in many cases, the lack of a degree (in someone who nonetheless achieves success in Academia) is a marker of genius. For example, the philosopher Saul Kripke famously did not earn a graduate degree in the field, but nonetheless was singularly important to the development of late 20th century anglophone philosophy. His lack of a PhD is integral to the mythos of Kripke as an autodidact genius.

Loving two fields is not a good reason to do two PhDs.
The work you do in a PhD is so specific and focused that I guarantee it will "ignore" most of mathematics and most of electrical engineering. Matt Might has a great illustration of this: PhD school in pictures
By the time you finish your current degree, you'll have a better idea of what specific area (i.e., subfield) of research you are most interested in (if you still want to do a PhD at that point). Then, decide whether it would be more at home in a mathematics or EE department, and apply to grad school accordingly.

Are you also interested in getting two jobs or having two families simultaneously? At some point in life people make decisions, long-term decisions that chart out a path for their lives.
The fact that you love both subjects is nothing but positive, and the fact that you have the drive and will-power to pursue higher education in both is admirable. It really is... But here I have lay out some points to consider:

Is it possible to do a PhD in two disparate fields simultaneously: probably.. I reckon it depends on the faculty/university and what they are willing to accept. Normally you can't get two PhDs from the same faculty (at least here it is not possible) and again, normally, universities do not accept other professional commitments while doing doctoral studies (with the exception of medical reasons). That is especially the case if you are officially employed as a grad student and not on stipends.
Is it advisable to go for two PhDs in disparate fields simultaneously: no, probably not. Doing a PhD alone is challenging, and I don't mean "OMG, it's too much work" but rather "OMG, wtf am I doing here!" or "OMG what am I gonna do with my life?!". Don't believe me? Just look at the most voted questions here, a healthy portion of them are about managing the mental burden of PhD studies.
Have you considered doing a cross-disciplinary PhD where you get to play with both subjects, meld them in a pot and make something awesome out of it? ;)

Hope it works out for you, best of luck!
PS - also worth reading: When does one go for a double doctorate?

+1 except for "Is it advisable to go for two PhDs in disparate fields simultaneously: no, probably not." I would say definitely, absolutely, 100% not. Nobody will fund you to do two simultaneous PhDs, nobody will agree to advise one of two simultaneous PhDs, nobody has the time or energy to do two simultaneous PhDs.

@DavidRicherby I full-heartedly agree with you, just didn't want to rule out the possibility of such a person existing out there. I mean there are... what, like 7 billion people out there? :)

Good point, though I'd describe a worse-odds-than-winning-the-lottery event as "almost certainly no" rather than "probably no". :-)

There may actually be quite a few people with time and energy to do two simultaneous PhDs, but seems to be no valid reason to do two simultaneous PhDs.  You do research in a narrow direction or a bunch of different directions. As a side effect of that research, you get a PhD in one discipline or the other, or a crossdisciplinary - it doesn't particularly matter, and you can choose. If you later believe that you should have had chosen otherwise, then you get another PhD in that discipline, but again that's more like a 'why not?' reason, and a side effect of research you already did anyways.

I'm on my second PhD, 40 years after the first one.  My first one was in mathematics, and I worked as a university teacher in maths for many years.  I agree with the general advice given here.  A PhD is as much research training as anything else.  In the course of it you work intensively in a specialised area and learn how to develop an achievable project that can be done in the time you've got. You will pretty much be living and dreaming your project. It would be very difficult to be so intensively involved in two completely unrelated projects at the same time.
In your case, see how your current course goes.  If you're still enthusiastic about both fields, look around various universities and see where there are interesting projects across both areas - aiming to do one PhD, and then maybe a post-doc position of some kind, if things work out that way.  I knew a guy who studied the applied maths/engineering of the very high-speed spinning machines used in the textile industry - weird instabilities could lead to disaster.  Another project I came across involved applying image processing techniques that had been developed for astronomy to classifying microscopic images of cells for medical research.  There are lots of interesting hybrid fields out there: the more engineering/physical side of robotics, remote sensing, aspects of fibre optics, the bionic ear and bionic eye - lots of innovative signal processing needed!
As for my second PhD, it is in fine art, but I'm writing computer programs to "evolve" artworks (not a new idea, but there's plenty of room for new work).  I need to engage with the history of art and other aspects of the discipline, since I'm in an art school now, though one if my supervisors is in computer science.

After writing my answer, I started wondering whether anyone had in real life done a PhD in both mathematics and art history.  So my eyes popped out when I saw this: it sounds very interesting!  May I ask how you arranged to do a PhD in fine art but with a CS supervisor?  That sounds...nontraditional, to say the least.

@PeteL.Clark: As far as I know Samuel Eilenberg only had one PhD (in mathematics, of course), but he was also a prominent expert on Asian art.  Someone once told me that he sometimes consulted with galleries and museums who knew him as "Professor Eilenberg from Columbia" without realizing he was a professor of mathematics.

@PeteL.Clark Replying to an ancient post because I was looking up "two PhDs" on here, but almost but not quite Tadashi Tokieda  https://en.wikipedia.org/wiki/Tadashi_Tokieda and

@MarkMeckes How weird. Eilenberg came up in my 'day job' recently. Odd to see the name here too. What a coincidence. https://www.newindianexpress.com/opinions/2022/mar/26/blood-antiquities-from-kashmir-in-a-new-york-museum-2434260.html

It's too early for all that - it is impossible to know now what you will enjoy by the end of your first degree, and there is no bonus for deciding early. I started my Physics degree loving space and looking forward to rocket science, by the end I wanted to do theoretical quantum physics, and now I do scientific/numerical computing. If you enjoy Maths and EE, you may end up enjoying a specific part of EE, like signal processing or quadrotor flight algorithms, etc.
You won't want to do a second PhD - A PhD is characterised by a lot of legwork, a lot of self-discovery and usually a capitulation to reality over hope in order to get something finished for the thesis. The important parts of that are not worth repeating. The only time a second PhD is worth doing is if you realise after completing one that you want to do research in another field, and that you can't make a path from one field to the other. A PhD is essentially the first step on a research path in a particular field, so make sure it's a field you want to do research in.
What happens after the PhD(s)? - A doctorate is just a step on a path to being a researcher, a lecturer, a professor. It is often the last piece of education people do just for the sake of a qualification. Afterward, if you go into research, you will have to choose a field, closely related to your PhD topic. You cannot go into two fields; the closest is to go into an interdisciplinary area between the two, hence quadrotor flight algorithms or sigproc. If you don't go into research, some experience of work would be a far better qualification for your CV than a second PhD.
My advice - Spend your current degree exploring the different fields. When you're intrigued by something in a course, follow it up with the lecturer, do some additional projects in your own time. Lecturers love people with enthusiasm and drive for a topic, and they will find time and resource for your project, perhaps with an eye to larger projects later. That process will be instructive, and help you understand what work you enjoy.
I realised I enjoyed numerical computing when I spent hundreds of hours expanding on a genetic algorithm project in my second year. Two years earlier, I wanted to build rockets. I would still like to build a rocket, but I would derive much of my enjoyment from simulation and design, rather than fabrication and testing; you will find that there are specific fields that match your skills and pique your interest, so accept that you will specialise, at least for a while.

"The important parts of that are not worth repeating." This is the best message. But it is a message that most students will not understand unless they have faced it themselves.

It's barely even a good idea doing one PhD, let alone two! What you need to decide is what career you want. If it's to be a researcher, then doing a PhD is appropriate, because it is training to be a researcher. Once you have that career, why do another one, unless moving to a vastly different research area that requires very different research methodologies?

Truthfully, there are few, if any, reputable universities that will allow you to pursue 2 PhDs at the same time. Also, most universities will not accept students who already have a PhD, especially when they overlap with one another.

To add to some of the existing answers, a desire to do two PhDs in different fields may be a sign that you want to combine your interest and skills in those two areas. This is very reasonable. There are also arguably several better alternatives for combining two fields in a research career than completing two separate PhDs.
The main option is to do a PhD in one field that links in with the other field. With regards to the linking, you can have a PhD supervisor who works at the intersection of the two fields, you can get a co-supervisor who works in the second field, you can self-teach yourself the other field, or you can do formal training in the other field.
To make this discussion concrete, I can point to many examples where a person combines skills in one field to complement another.

I personally completed a PhD in psychology, but have endeavoured to acquire skills in statistics (through courses, self-teaching, a Post-Doc with a statistician, etc.).
I know a student who has a law degree and is completing a PhD in forensic psychology.
I knew a professor who had published books on music, and studied the psychology of music.

If after your undergraduate studies you find that you still enjoy engineering and mathematics, you may be able to find a PhD in one of these fields that links with the other.

An important difference between a bachelor's degree and a PhD is this: when doing a PhD, there is no maximum amount of effort one can usefully put in, nor is there a maximum level of attainment one can achieve. (Whereas with a bachelor's, once you have got top grades in all your courses, there's not much more you can usefully do.)
Supposing, what is very unlikely, that you have the academic ability and the huge amount of motivation and energy it would take to do two PhDs simultaneously. Then instead you should do one PhD, but:

in a shorter time than is usual
writing an astoundingly good thesis
publishing significant work
which you have largely done on your own
and having gained an exceptional depth of knowledge about your chosen field

For you to do all or most of this will be no more work than two simultaneous PhDs, but it will get you a lot further. Graduate students who do PhDs like this (they are quite a few) tend to quickly build prestigious and desirable careers for themselves, whereas 'professional students' who spend their lives amassing diplomas in different fields usually don't.

Well, I don't think its everybody's goal to get desirable careers. I know some people who do PhDs beside their works not because they need them, but just because they are interested in those areas, but not enough interested to leave their work and family to do a postdoc somewhere else

You can not sail in two boats, a foot in each ?
First make up your mind about GOALs in life,  what are your objectives .
See how best to  get them.
PhD  for what purpose ,  NOT just for sake of  having on Name plate outside house or   for filling CV.
Worry not, you have enough time to take that decision,  but  Decide about your GOALs today,  you will have a better life, well managed.

Going 100% for two subjects might be hard.
One way, which might however be an opportunity for you, is to write an electrical engineering PhD that is very mathematical. Vice verse there are certainly many mathematical problems that arise from electricity question.
If you do this you will surely be able to study at both fields. Also, no matter what field you would choose, you would need to make sure that your thesis really belong to that field.

Definitely not. Try to concentrate on one first and do the other as a hobby. You will get a lot better because you don't have double-pressure. Alternatively, when you finished the first doctor degree work for several years (I'd say 10-20) and then you can possibly start with the other doctor degree.

In most cases you simply will not get offered the second PhD, even if you apply, and this remains true also if you try to do these two PhD positions in different countries. I am not aware if these are written rules, or unwritten rules, but, really, mostly no avail.
Unlike master studies, PhD is often supplemented by stipend that, while small in general, is enough to sustain a living. Probably because of that there are some efforts to avoid a "professional PhD student" who would keep "studying" till retirement.

It is essentially impossible to do two PhD's concurrently, but if you like you can begin a second PhD after completing the first.  Keep in mind that many people drop out of (even the first!) PhD, and are generally quite drained by the time the conclusion arrives.

The answer to your question depends on your goal.
In my experience in the software industry, if your goal is to go into industry, a PhD often gives the impression of someone who is "overly academic". Someone who holds a doctorate may be seen as someone who preferred to stay in the comfort of academia. I imagine someone who had done two PhDs would be practically unemployable in the eyes of many hiring managers.
If your goal is to work in industry: I'd advise dedicating time to locating a company with highly skilled teams doing work that really interests you. Build relationships with those people and go work on those teams. I imagine working as a team on a real world project might be more enjoyable and inspiring than a PhD. You'll probably learn faster and broader too.
I'd advise observing the general demeanor of people doing PhDs too...

I think suggesting that PhDs are practically unemployable in industry is going too far. There are many opportunities for PhDs in industry and some industrial research positions require one. At least, it may depend on the exact field and country.

I have been in industry and other non-academic organizations with (one) PhD.  It is not really an advantage for most jobs; it was for at least one.

I have a PhD in Computational Biology. I have worked in that field, various engineering fields, am currently in management at a large corporation, and have even worked in finance. The point being, that once you have a PhD in a technical field, you generally have the credibility to switch fields to a number of other technical fields. It certainly takes some work to do so, but quite a bit less work that earning a second PhD.
Furthermore, about 80% of the process of getting a PhD is learning how to research and the other 20% is learning subject matter. This carries a couple different consequences in regard to pursuing a second PhD. First, while you could finish a second degree much faster than a first, there is a certain amount of structure to graduate school and you will invariably waste time developing (or rather proving that you have already developed) skills that you learned the first time around. Second, once you have the research background that a PhD confers, you can quickly learn the subject matter required to conduct novel research in a relatively similar field - it's not easy, but it's not that hard either.

I do suggest completing your Bachelors first to see how you feel about progressing onto postgraduate studies. Having said that, I'm an individual who is completing two Masters degrees simultaneously. It may seem like an impossible task, but really it isn't.
Just like you I love both fields and really didn't think I should choose one over the other. I also work full-time (Executive position), and lead a pretty full life with sports, yoga, meditation, etc. It's all about knowing how to organize your time and still maintain balance.
So, if at the end of your Bachelors you feel you can do it, go for it. Why not? It's your life. The key is try to balance your life. You don't want to find yourself coming to a point where all you do is work and have no play. Then again, if you love your fields that much, work can be more like play. This isn't an impossible task at all. Passion for what you do, coupled with focus and organization are the key to your success. Really. Take it from me.
At the end of the day, do what feels right to you. You know what's best for you, not anyone else.

First of all, in your case, it should be obvious that it is way, way too soon to even think about a PhD of any kind.  In many ways even one PhD is too much these days - given the employment market today, opportunities for PhDs are slim to none.  Best advice? Keep in mind that having a PhD often closes more doors than it opens.

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text:
Does graduate school reputation have any effect on your academic career?
I am an Australian citizen and studying at University of Newcastle (Australia). Does your graduate school reputation has any effect in your academic career? So if I decide to do my PhD at a US university (like Harvard or Stanford) instead of doing it at an Australian university (say University of Melbourne or Australian National University), would that make me more successful in my academic career? Does that make me more employable? Say that after doing my PhD, I apply for an academic position at a Canadian university, which of them would help here more?

This seems like a duplicate of http://academia.stackexchange.com/questions/90/university-rank-stature-how-much-does-it-affect-ones-career-post-ph-d/154 This question is a bit more specific, but I don't think that will change the answers.

The short answer is yes, of course.
The long answer is, it depends. Ultimately, your future career will depend on your publications and other output, their quality and their impact. If you can produce a number of high quality publications from a lower-ranked university, this will be better than producing nothing of interest from a higher-ranked university. However, the chances of "succeeding" are higher at higher-ranked universities, because they have "better" people to help you along the way. They also have higher standards and more pressure. So there will be a bit of a trade-off.
That said, the University of Melbourne and ANU are good places.

I'm glad you put "better" in quotation marks.

Here's another way of thinking about this. Yes, of course, at the end of the day what really matters is your scholarly impact--lots of good publications overcomes a multitude of sins. However, the thing to realize is that you are much more likely to have the time and resources to get those publications, the better your first academic job is. If you get a big travel budget and a lighter teaching load, you just will be able to publish more than colleagues who get jobs with heavy teaching responsibilities. And to get that first job, the prestige of your PhD is one of the primary factors.

@shane: why not make your comment an answer to the original question?


text:
What is the general protocol to name something after somebody (some people)?
In mathematics (and other sciences) there are thousands of concepts, theorems, lemmas, etc which are named after some mathematicians (scientists). However, this nominations are not always very straightforward, especially if we are going to assign a new name for new concepts. For example, I can imagine the following scenarios and I would like to know what general protocol we should apply in each case:

A concept may have several origins in different fields and due to several individuals.
A concept is built on another concept which already have a name and this can happen several times. For example "Hecke pairs" is a concept in mathematics, then Bost and Connes make a particular Hecke pair famous, so we have "the Bost-Connes Hecke pairs". Should we name all influential people  in each stage of advancement of a concept?
If an author invents a concept, is it appropriate to name it after him/herself,  or he/she should wait others call it after his/her name?
A concept was invented by some author "X" in long time ago, and then it has evolved to something very modern and somehow different. Should we still call it by the name "X"?

Please do not hesitate to add new items if you can imagine other scenarios too.
Finally, I would like to ask another related question:
Can we use acronyms in stead of the full names, especially if the names of several people are involved?

A concept is usually named for the first person to "discover" it.  For example, the sequence discovered by Morse was called the Morse sequence.  Then it was found out that it was discovered earlier by Thue.  So they called it the Thue-Morse sequence.  Then it was found out that it was discovered earlier by Prouhet.  So it should be called the Prouhet-Thue-Morse sequence.  (But for some reason I still see it called the Thue-Morse sequence.)

About item 3, I should add that I am not asking this item because I want to name something after myself. It is just a possible scenario which I thought that it should be considered in my question.

Relevant link on how diseases are named: http://www.nytimes.com/1991/06/02/magazine/on-language-name-that-disease.html

Good questions.  I will only tackle the last two:
3) In mathematics, it is (virtually?) universally bad form to name something after yourself.  This is high on the list of things that amateurs/newbies do that make the professionals/veterans roll their eyes.
Some people have joked that the best strategy to get something named after yourself is to give your nice new concept such a terrible name (or lack of a name) that the rest of the community converges on naming it after you.
Even after something has been named after you, it is not necessarily completely kosher to speak your own name when referring to the concept.  Rob Kirby famously speaks of "the calculus of framed links" (or, I think ironically, "the calculus") where others speak of "the Kirby calculus".  At one point Armand Borel writes of "the subgroup whose name I have the honor to bear".
It gets a bit ridiculous: when you give a talk and state one of your own theorems, it is most common to write out the names of your coauthors and not write out your own name.  In my student days I saw a lot of first letter then dashes.  Nowadays I mostly see the name dashed out entirely.  Come to think of it, this reminds me of the Jewish practice of leaving letters out of the name "Jehovah", although the theological implications of treating your own name this way are much more profound.
4) I don't know whether we "should", but we often do. In general, it seems to me that mathematics has gotten used to naming things after certain people, and we often name things after people who would never have understood the things that are named after them.  The example Hilbert space (coined by von Neumann in its present generality) is a famous one.  The example Euler system has always struck me as being especially ridiculous (I asked my advisor about this, and he told me that the name comes from Euler products: that's quite a stretch).
Some people in mathematics are somehow especially good at getting things named after them.  In my field, perhaps the outstanding example is John Tate: he has curves, algebras, half of the Shafarevich-Tate group, half of Hodge-Tate weights, half of Lubin-Tate formal groups, a pairing...As a graduate student, I was struck by the fact that I was giving a talk on Galois cohomology of products of Tate curves, analyzed via Tate local duality.  The title of the talk, "Tate-Tate-Tate Stuff" was a riff on the title of the previous speaker's talk ("Hodge-Tate Stuff") and this Tate-ish ubiquity.  When Tate showed up for the talk, I got very nervous...but he was cool with it.
Needless to say, John Tate is a true luminary.  The fact that so many things bear his name is only possible because of the immense amount of fundamental work that he did.  But the converse does not hold: e.g. Barry Mazur is a mathematician with a similar impact on the field, but he has...what?  A manifold and a swindle?  (Both of these come from his work in topology at the beginning of his career.)   Instead we have the Eisenstein ideal.  These things are strange.

And Craig Tracy refers to the Tracy-Widom distribution as simply $F_2$, the notation used in their original paper.  (I attended a talk of his once and was rather nonplussed until a colleague whispered to me "He means the Tracy-Widom distribution").

+1 So one should be fashionably modest when presenting your own theorem? ;-)

I agree about your points, but I have seen Paul Baum talking about the Baum-Connes conjecture and it looks inevitable somehow.

@Vahid: Yes, I definitely tried to indicate that the practice of refusing to speak one's own name in a mathematical context is slightly weird.  On the other hand, phenomenologically speaking: yes, we can be weird...but are much more rarely required to be.  You really shouldn't name something after yourself in mathematics.  Other than that I'm just telling stories.

For example see this notorious article http://care.diabetesjournals.org/content/17/2/152.full.pdf+html in which a medical researcher essentially publishes the trapezoidal rule in 1994.  To me one of the strangest and most cringeworthy aspects is that throughout an article written by Mary M. Tai she consistently refers to Tai's model.  From this I deduce that this behavior is entirely reasonable in some academic circles and would be interested to hear more about that.  In a math journal that practice alone would probably prevent her paper from getting published.

Tai's model led to some rather heated exchanges in the letters column of the journal: see http://www.math.uconn.edu/~kconrad/math1132s14/handouts/taicomments.pdf for a collection.  In my opinion, Tai's explanation probably makes things worse.  It contains the relevant passage "Besides, if I do not address the model as 'Tai's', other investigators who wish to cite it will."

In some fields (such as mathematics and physics), naming something after yourself is a sign of a crank.  (See Number 25 in John Baez's "The Crackpot Index".)

Your fourth point implies Hilbert would never understand what a Hilbert space is!!!!!

In order to disabuse you of the idea that there's a reliable system to name things after people, I present to you: Stigler's law of eponymy.
And if that's not enough, you'll occasionally have item A invented by author X but named after Y, and item B invented by author Y and named after X.
My usual joke about this is that something is usually named after the last person to invent it, because they're the one to popularize it enough that no one else can reinvent it.

Related to Stigler's law is Noche's Principle: "Never name a concept after yourself."

Like Baumslag-Solitar groups were appeared first in the work of Graham Himan in 1950, but named after Baumslag-Solitar.

"something is usually named after the last person to invent it, because they're the one to popularize it enough that no one else can reinvent it." Side-question: do you consider this good or bad? Because it seems to me that popularizing an idea to the extend that it becomes general knowledge is in fact often the harder contribution, and requires a lot of "going the extra mile".

I don't judge the popularizer vs the creator. To me it's a (yet another) cautionary tale about the fickleness of fame and recognition.

The following has been my impression as a biologist:

What is the general protocol to name something after somebody (some people)?

There is none. For certain things, such as names of genes or species, there is a protocol for submitting a name to the relevant databases, which is a right reserved for authors of the publication. This name can be anything you want, although certain standards are encouraged.
People often name species after their own name. For plasmids, the convention is to acronymize the name(s) of people who created the plasmid and make it the name. For genes, this would be considered tacky (the fashion seems to be to naming them after "clever" puns instead) but I'm sure you could get it to happen with enough perseverance. But there is nothing special about it being your name, because the name of the thing is arbitrary. You are allowed to give it any sort of name, your own name is just one of the (less interesting) options.

A concept may have several origins in different fields and due to several individuals.
A concept is built on another concept which already have a name and this can happen several times.

Concepts are not formally named after people. When originally published, the authors may or may not invent a term for the concept they discovered, to facilitate its discussion. It gets named after them, when the research turns out to be so seminal that everyone cites and recites it, and the authors begin using "the Smith protocol" as shorthand for "protocol described in a recent high-profile publication by Smith and colleagues (Smith et al. Nature 2012)". If it yet persists the test of time further, it may become a de facto tradition to call this the "Smith protocol", especially once textbook authors start electing to use "Smith protocol" as the canonical name in their own texts.

If an author invents a concept, is it appropriate to name it after him/herself, or he/she should wait others call it after his/her name?

The exception I name earlier notwithstanding, absolutely not. A scientist would get laughed out of the room if he tried to present something he blatantly named after himself (some subtle reference to his name, like an anagram of his first name, might be begrudgingly accepted), unless he was perhaps a famous Nobel laureate.
If he was a famous Nobel laureate, people would still laugh, they would just wait for him to leave the room first.

A concept was invented by some author "X" in long time ago, and then it has evolved to something very modern and somehow different. Should we still call it by the name "X"?

Since I assert the naming of concepts happens not through formal procedure, but as a consequence of frequent references to the original publication, then an improved "Smith protocol" may be named the "Doe-Smith protocol" or "Doe protocol (based on the Smith protocol)" or even just "Doe protocol" if Doe manages to publish a paper which provides a useful reference for the improved version, and the improvements are substantial enough that people feel the need to cite and refer to Doe's paper at least as much as Smith's paper.
If you were trying to get something named after you, the realistic strategies in biology are:

Discover and name a new species, plasmid, gene, etc. And hope the nomenclature committee doesn't think you're being too arrogant.
Describe a new experimental or mathematical/computational method, and fail to give it a nice name yourself.
Write a definitive reference which synthesizes several existing ad-hoc variants of a concept into one unified theory, and fail to give it a nice name yourself.

For 1, formal procedures exist and are detailed by the agency you submit your name proposal to. For 2 and 3, you basically write the paper, and wait for everyone and their brother to start citing your landmark publication - hopefully they will talk about your research so much that the name you used will prove too cumbersome.
Some examples:

The famous Southern blot is described as only "a method of transferring fragments of DNA from agarose gels to cellulose nitrate filters" in Southern, 1975. Although extensions of this method, like the Western, were important discoveries, their popularizers got a bit less glory since it turned out that geographical puns were more fun.
Eagle's minimal essential medium is described as "a fluid medium" in Eagle, 1955.
Okazaki fragments were not referred to as such in Sakabe, Okazaki, 1966.


text:
Lazy/Busy Professor does not give me my confirmation
I have done a short summer research internship at a department of a famous University in Europe. Unfortunately, this was last summer and I reminded my professor several times now to give me some sort of written confirmation that I actually stayed at his department.(3 times per E-Mail and one time, when I left him). He always said that he will do it, but he always pointed out that he is very busy currently.
Do you think this sounds true? I mean, it has been half a year now and apparently there is not much to do about it instead of just waiting or is there? I also just asked for a few lines, not a confirmation letter of anything similar and I also told him that I would need this for my home university(Which was true)- but he did not really react upon that.
I do not want to pressure him by being more "rude" in my mails, this is not the way I deal with such situation, but I think his behaviour is very annoying and I want this piece of paper now.
How would you deal with that?

I would address this by contacting the professor's administrative assistant or secretary. Usually, such "form letters" do not need to be actively written by the professor in question—just signed by the professor. The assistant can prepare the letter, and get the faculty member's signature; in some cases, the assistant may even have a digital signature available, so the professor's direct involvement isn't even necessary.

A similar option would be for the OP to draft such a letter themselves that could be printed on letter-head and signed. (Even if it were not used, it would have the effect of guilting the professor into action.)

Note that you may not even need to contact the professor's office at all; if you were a student in a graduate department, a letter from the department office or department chair may suffice.

This is a good approach in general: only ask the busy person if you actually need the busy person, otherwise be creative in finding someone near that person who can handle it.

Also a take-home message: get all your proof documents (whether you needed them at that moment or not) while you're still physically there.

Most professors really are pretty busy. So, if you want something from them, make it as easy as possible for them to fulfil your request while staying respectful. In a case like your's, prepare a letter and ask them to sign it (or to copy the text electronically in their own letter template).

Hah! It took me 4 years to get a confirmation for my exchange student year to get my degree. After numerous emails and even official letters from my University which all got ignored, the only thing that finally worked was a flight back and running around there for a week, where everyone was telling me they had no trace of my stay whatsoever and putting the responsibility on someone else. So I hope you don't need to do those extreme measures but stay prepared and first go to your departement and let them write a formal letter. Also phone calls may work better than emails. Good luck!


text:
How to deal with procrastination and delays in final assignment of undergraduate class?
This semester I taught an optional course for undergraduate students. While I find it normal that fewer students attend optional classes, presence in class itself was low even for the enrolled students, and now the only requirements for evaluation have not been met by most of the students. Only 7 out of a total of 21 students have turned in their final written essays until the deadline.
The problem is, the university grading system allows for students who have not passed the exams during the semester to take a "recovery" exam/written assignment. Through this loophole, students who have not even written their essays may pass if they get good grades in this "recovery" exam. While I think it is somehow unfair for the other students, because it allows procrastination and delays of over a week, and gives a second chance for students who did poorly during the whole semester no matter what the reason, it is not clear if I am supposed to apply this "recovery" exam or if this is optional.
The question is, if I do not do this (accept procrastination and delays), most students will not pass.
Should I just accept it or be strict with rules and deadlines?

It would be completely reasonable — if you announce it at the beginning of the term — to make the recovery exam/assignment available only to students who have demonstrated a good faith effort by turning in all regular assignments.

Wouldn't it be better to check the rules for the recovery exam with the department/university? They surely know the answer; we surely don't.

Your question gave the impression that the only evaluation is the final written assignment.  Is this correct?

If the students were reasonably expecting the option of a recovery exam, it seems unfair to deny it for the current run.
In future, you could offer the recovery exam only to those who tried to pass in the regular way, i.e. handed in an essay on topic (but maybe bad and hastily written - so more unpleasant grading for you).
Generally, I would suggest talking to some students to figure out why they didn't hand in an essay. Maybe it was plain procrastenation, maybe it was wise prioritizing because they had other, more important deadlines in the way, maybe they found the course not that interesting and decided not to complete it. In my experience, undergrads often have more self-awareness of their learning process than we would expect :)


text:
Taking higher level theoretical mathematics vs taking less math and more courses in application areas?
I intend to go to graduate school for applied/computational mathematics, specifically a program like this https://icme.stanford.edu/.
At this point, I'm trying to decide whether to take graduate level theoretical math courses in areas like Algebraic Topology, Differential Geometry, etc. (which I don't currently have any experience with, but I could still take), or just take courses in application areas (the ones I'm interested in are statistics, biology, chemistry, computer science).
My math education in the latter case will consist of basic linear algebra+calculus, a course on PDEs, a course on abstract algebra, a complex analysis course with an applied focus, a couple mathematical modeling courses, basic number theory, basic probability theory, a couple numerical analysis courses, and upper-level real analysis. So not a ton, but not negligible either.
There's also a lot of math in the non-math courses I will take (algorithms, theory of computing, quantum mechanics, optimization, stochastic processes, machine learning, etc., some of which are grad-level) However, this schedule is perhaps lighter on mathematical theory, and contains zero grad math classes.
Should I drop some of the courses in application areas (although courses with significant mathematical content that yet focus on applications are perhaps my favorite type of courses) and take graduate-level theory courses to increase my readiness for and chances of getting accepted to graduate school? Or will I have ample time for that in graduate school and should I take courses I enjoy more (and have greater aptitude for) while informally studying theory on my own?
Or do I need to take more theory even as an undergrad, and even though I'm not gunning for pure maths? (I do not plan to go into academia after graduate school, in case that matters.)

I think that at this stage in your career, you should set yourself the goal of reaching some sort of mathematical maturity

... fearlessness in the face of symbols: the ability to read and
  understand notation, to introduce clear and useful notation when
  appropriate (and not otherwise!), and a general facility of expression
  in the terse—but crisp and exact—language that mathematicians use to
  communicate ideas.

Courses like abstract algebra, algebraic topology, differential geometry and e.g. combinatorics will give you a combination of breadth and depth that will make it much easier to quickly master any applied field that you will choose in grad school.
In my experience (PhD in string theory, currently applied economist), mastering the really advanced math courses should be done as soon as possible, when you have all the time and energy to immerse yourself. It's also my experience that it is rather straightforward to apply abstract patterns when you already know them, but the reverse (the emergence of abstractions from concrete applications) is much, much harder.
Of course, by all means mix and match your fundamental math courses with a small selection of interesting applied courses. There's no sense in not enjoying yourself.

I would take the applied maths courses (the second option). If that is what you enjoy and would like to pursue in grad school then it will benefit you more than theoretical courses.
I take more theoretical courses to mean a lot of epsilons and deltas and theorems etc. If you are more interested in statistics, biology, chemistry and computer science then these courses will not be as useful as more applied mathematics courses. In applied mathematics you're more interested in constructing interesting algorithms or models that work without worrying about the theoretical details too much. That's not to suggest that applied mathematics is any easier, it's just that the focus is different.
Do you have any specific ideas what area you'd like to work/study in?

I'm not specifically sure, but most likely something in the set [chemistry, molecular biology/genetics, bioengineering, chemical engineering, materials science] with a theoretical/computational focus. After grad school, I probably want to go into quantitative finance unless I can get a good job in industry that is similar to what I do in graduate school.

This is really quite a subjective question. However, even if you are planning to work in applied areas, it probably makes sense to get a good grounding in basic theory first. For example, you say you are interested in statistics. If so, you should definitely take at least one advanced probability course which uses measure theory, and also a measure theory course. These should definitely help later on, even if you don't wind up using advanced probability theory. Some material at the level of the Billingsley book "Probability and Measure" is kind of what I am thinking of.
More strictly pure courses like Algebraic Topology, Differential Geometry are a bit more debatable. They might be useful in applied areas, depending on what you are doing, but will probably not be. They might be worth it from a mental broadening perspective, but that is really subjective. I've worked in applied areas some, and have never needed to reference theory of this kind.
Also, I think if a course is well taught (which may not be the case, of course) and forces one to work on the material, then it is better than self study.
Disclaimer: I have a PhD in Statistics, which may cause some biases.

OK, I'll probably just save topology and geometry for enrichment in grad school, if I have time (my advisor did recommend geometry, but he said it's not a huge priority.) As for probability theory, it's not going to be my main area of focus, but it is an area I think is interesting. Do you think it would be a good idea to skip the undergraduate probability course and go straight to a measure theoretic probability course? Or should I try to take undergrad probability and measure theory separately? Or even undergrad probability and a grad stochastics class?

@user2258552 I think an undergrad probability course + a grad probability course using measure theory is good. If you are trying to learn basic probability + absorb measure theoretic probability notions at the same time, that would be too much for one course. You can and should (imo) take a separate measure theory course. As long as you take it  before the measure-theoretic probability course, that would be fine. That would just be a math course. I wouldn't try learning measure theory in the context of a measure-theoretic probability course, because again it would be too much.

Just to be clear, taking these courses if you are working with probability + statistics isn't some kind of intellectual luxury or cultural broadening exercise; they will inform the way you think about these topics. Far too many people work in statistics without even a basic mathematical grounding of the kind I am advocating. Also, no offense, but you really should be discussing this kind of stuff with faculty mentosr who know you (not limited to your advisor, necessarily), not with strangers on the net. Doesn't your university have these kinds of support services?

I know it's not a intellectual luxury, but I'm not sure that I will actually be working with probability and statistics heavily, and if I do I can take the class in grad school (as they are graduate-level anyway). I'm just trying to figure out which ones are most crucial.
As for your other comment, yes I do discuss with my advisor, he is very helpful, and I am going to meet with him again, but I don't want to keep emailing him, and the more people's input the better. It's not like nobody on this site is qualified, much the opposite, and I can weight different people's opinions as I see fit.

Also, this is not exactly sensitive information, you would probably have to know me personally to guess who I was just from reading this post, and even if you did, I wouldn't care, so I'm not sure why I shouldn't discuss with 'strangers on the net.' But thanks anyway for your input.

@user2258552 Ok, I agree that your perspective sounds reasonable. I'm not suggesting that asking here is a bad idea, obviously not, just not something to rely heavily on. The point is not that the information is sensitive, just that advice from strangers who don't know you is inherently limited in its usefulness.

For some applications of Algebraic Topology, see the work of people such as Robert Ghrist and Gunnar Carlsson.