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In (pure?) mathematics, author order on papers is alphabetical, based on the assumption that all authors contribute equally (e.g. as discussed here and explained in this statement from the AMS). Moreover, single author papers are quite common, even from students. Many math PhD students will have few (or no) papers coauthored with their advisor, and when people from the same research group collaborate, the usual practice of alphabetic authorship is followed.

As an outsider from a "related" field (theoretical physics), I'm wondering how this works in practice and what the experience of a PhD student in mathematics is like. The description of "joint work" from the AMS statement makes sense in describing collaboration between colleagues, but not advisor-student collaborations.

In physics, typically a professor will suggest an idea, and the student(s) will explore the idea, do some calculations, etc. Then they'll have some discussions, the student(s) will present their results, the professor offers some suggestions, the student(s) do more calculations, and eventually they write a paper.

How does it work in mathematics? Is it similar to physics, where the advisor offers ideas and direction, but with the convention that the student typically writes up the results as a single-author paper? Or is it the case that mathematics PhD students really are expected to be highly independent, and receive much less help from their advisors compared to graduate students in other fields?

A related question of mine is whether there is any "grunt work", so to speak, in mathematics research. There are no experiments to do or simulations/computations to run, but is there still a certain amount of calculations or proofs that merely require "routine" work?

For example, I imagine that mathematical research, conducted "physics-style" might look like this: Professor thinks of some theorem to prove, problem to solve, etc, and introduces it to a student. The student works on it for while, and then brings their results to the professor, who then offers their interpretation and new ideas, and so on. Over time, they "jointly" work on the project, but the student does the vast majority of the pencil-on-paper work. Is this style of research (or rather, of advising students) not common in mathematics?

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As a preface, which might help explain the phenomena you observe, "grunt work" tends to have low status in mathematics. A solution to a problem that uses a large amount of "grunt work" and little else might be unpublishable, while a different solution (providing the same answer) to a problem that involves much less "grunt work" might be publishable in a fairly highly regarded journal. Remember that results in mathematics tend to have less (but not no) extrinsic motivation, so part of the evaluation of a result is the elegance with which it is achieved. (This general valuation extends to entire subfields - graph theory is a low status field in large part because proofs in graph theory tend to have a high grunt-work-to-insight ratio - for example the proof of the Four Color Theorem involves a fair amount of insight but a gargantuan amount of grunt work, so much that it had to be carried out by computer.)

It's worth noting in this context that a significant fraction of dissertations in math are never published as papers, because the results in them (or rather the manner in which they are achieved) aren't considered interesting enough to be published in a reputable journal.

So - a dissertation in which an advisor provides a problem and the student solves the problem by pure grunt work is unpublishable, and terrible for the career of the student. Also, if you are looking at this from the outside using the journal publication record, you get a misleading view because the dissertations written in this way are simply never published as papers in journals.

This doesn't mean a student is expected to be completely independent as might be the case in many (other) humanities fields. There is indeed a cultural norm where many (probably most) mathematicians do not put their names on work of their students even where, if there was not a student-advisor relationship, their contribution would merit co-authorship. It's quite common for advisors to offer direction and hints at ideas, but the student writes up the work as a solo paper (with an acknowledgement for the help of their advisor). In general (though not always, and each research community tends to know who the exceptions are), when a paper is jointly authored by an advisor and a student, that actually means the advisor did all the important work and the student contributed only grunt work and no insight.

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    Thanks, that is interesting. In this sense math is very different from physics (and probably most fields), where the advisor has a much lower bar for authorship. Example: a potential project arises in a discussion between A and B. B is too busy or not interested enough, so A pursues the project. A and B have a few discussions later on, as A writes the paper. In physics, I would imagine that B might (though not necessarily will) be on the paper if he were A's advisor, but maybe not if otherwise.
    – Aqualone
    Dec 3, 2023 at 0:25
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    I really do think the cultural prevalence of whether advisors tend to put their name of students' papers is so unclear as to render a speculative "most" very ambitious.
    – user176372
    Dec 3, 2023 at 2:33
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    Hnm, I'm wondering a bit about the sentence "It's worth noting in this context that a significant fraction of dissertations in math are never published as papers." I think almost every PhD student I've got to know during the last 10 years or so (in mathematical analysis) published at least a paper or two. Of course usually not in super fancy top reputation journals, but certainly in reasonable journals where people will typically assume to find serious mathematics. Dec 3, 2023 at 10:43
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    @JochenGlueck The statistics in this answer, averaged over different math fields, indicates a substantial fraction (roughly 50%) are not published or accepted at the time of an exit survey. I suppose the asymptotic value as t→∞ must be lower, but seems unlikely to be negligible.
    – Anyon
    Dec 3, 2023 at 16:03
  • @JochenGlueck: I work in a department where, apparently, in the entire 50+ history of our PhD program, no pure math student has ever gone on to a research-oriented academic position (not even a postdoc). That may skew my perception (but in more than half of math PhD programs in the US, less than one student a year goes on to a research-oriented academic position). Dec 3, 2023 at 16:20
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Just as another anecdotal data-point: (in the U.S., in math, at an R1) I view my job as PhD thesis advisor to be that of coach/mentor/educator/protector.

The "protector" part of my responsibility is to shield my students from working on things that I myself know full well I could not accomplish (despite being fairly clever, having decades of experience, and reading lots of other peoples' work).

So I suggest/encourage projects that I'm fairly confident will work out, and will be interesting, but ... hopefully ... are not so immediately on other peoples' radar that more experienced people will "scoop" my students.

Then I coach my students in such projects. Yes, I do give lots of specific technical commentary... and editing advice on documents ... but my students do the literal work.

On another hand, no, definitely, there is no "grunt work" that I need done to further my own research projects. I suppose this is wildly different from more lab-oriented things. My own main benefit from coaching beginners is motivation to re-think things myself. A large benefit, in the end. :)

I do understand that there are wildly varying authorship criteria... I take the viewpoint that I am a more senior person who happens to be still-alive, giving advice in person, as opposed to the many good mathematicians who've written useful things, but now can only communicate by their writings. I do make a point of citing historically significant (and, stimulating/helpful-to-me) documents... but I hesitate to give co-authorship to deceased people, no matter how much they've contributed to my thinking. :)

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    So, in the situation you describe (you suggesting the topic, and offering advice and editing) your contribution would not warrant authorship? I guess that really is a difference between fields, because in physics the advisor would definitely be an author even when the students did the "literal work" (and that's where the author order comes in)
    – Aqualone
    Dec 3, 2023 at 0:21
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    @Aqualone, yes, I guess it's a cultural difference... I think that, in math, everyone knows that the advisor (at best!) would indeed be very helpful to their students, but not claim co-authorship. Nowadays, yes, I do see some people claiming co-authorship with their students on thesis-related work. Understandable, I guess... Dec 3, 2023 at 0:35
  • Your last paragraph is interesting. In biomedical, I think the distinction would be published versus unpublished work. You wouldn't give coauthorship to someone for contributions which were already published in some fashion (you'd cite instead), but you would (potentially) for contributions which couldn't be found by independent third parties. Posthumous authorship isn't common, but is completely warranted if they've provided a unique (unpublished) contribution to the work.
    – R.M.
    Dec 4, 2023 at 13:15
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In pure math and some parts of CS there is a parallel, but the "do some calculations" part is, perhaps, more challenging than you describe. If an advisor suggests a problem (not a calculation problem, but a question as to the potential truth of some idea) then the student needs to do a lot of work to prove or disprove the hypothesis. It is non trivial. Along the way, there may be consultations and suggestions made. But if the student is able to provide the crux of the solution then it is usually the case that the advisor will let them take sole credit, though people understand that they didn't work in a vacuum as an independent researcher.

The opposite case can also occur, though less frequently in the student-advisor relationship. One mathematician can be working in a problem long and hard without getting the main insight (the crux). A five minute conversation with a colleague might provide that insight, in which case co-authorship is usually given.

Advisors (good ones) should be able to steer students away from dead-ends. This takes a lot of experience to have a sense about what is do-able with the current state of the art in a field and what is probably beyond reach in a year or so if deep thought. I worked on three different problems. The first (if I remember correctly) was too easy and I got results every day. It was "cute" but didn't have enough substance. The second was too hard and a month of work left me with no insight at all. It seemed intractable at the time. The third was just right. Hard and deep, but progress could be made building the needed parts and putting them together.

So, your last paragraph seems pretty close to the mark, but it isn't "pencil and paper" work. It is deep-thought and a search for insight that can be captured in proofs of meaningful statements (theorems).

I think it is often the case that the professor giving out an idea for a problem could do it on their own and may already even know how. But they leave it to the student to work through to the insight needed. I suspected that this was the case with my dissertation, as my advisor was a superstar. But he let me thrash through to the required end and it was a good result. It was my work, but he wasn't surprised as to where I got to. It wasn't a "collaboration", so I was sole author.

The reason for alphabetical listings when there is a collaboration is that the insight might, perhaps often, result from joint discussions where the crux emerges from the conversation and it is impossible to assign priority. These are true collaborations on ideas, not just dividing up a bunch of work into independent chunks. Collaboration in math is often just a search for "attack vectors" that can be applied to crack a thorny problem. One goes at it from many angles but discussions can provide new angles that may prove fruitful or not.

I would guess that in theoretical physics it could work the same, since it is often insight (the crux) that is the key and an advisor could sit back letting the student get there with some general guidance to keep them from going off the rails. I've read a discussion of Einstein and Special Relativity and the ten year quest for insight. That is a lot like what goes on in math. Not "calculations" or "pencil and paper stuff", but key insight and finding ways to express it formally.

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    As a theoretical physics professor, but with a Ph.D. applied mathematics, I can say that the fields are pretty different with respect to the degree of importance of insight versus calculation. I have had papers with graduate students where the students were first authors, because they did the majority of the work, even though the deeper insights may have come almost entirely from me. It's not an easy problem to figure out the right angle to take to solve a hard physics problem, but it is often even more work (and not intellectually trivial work, either) to apply that method and get the answer.
    – Buzz
    Dec 3, 2023 at 2:56
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    @Buzz: As I hint at in my answer, I don't actually think that mathematical problems inherently require a higher insight-to-calculation ratio - it's just that, in mathematics, problems that are solved by a lot of calculation are devalued because a lot of calculation is done to solve the problem. Dec 3, 2023 at 7:16

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