What is the primary means of education for doctoral students of mathematics? I’m guessing it’s seminars, not lectures – and monographs and journal articles, not textbooks. Is that correct? Are there many textbooks for doctoral students in pure math? A good friend with his PhD in engineering said he spent most of his doctoral work in labs, but maybe that’s because it was engineering. Or, is it that by time a student is working toward his/her doctorate, most of their time is focused on their own research and dissertation?
No doctorate, in any field, is primarily, or even much at all, about textbooks. A PhD student will be reading journal articles, and talking about them with their advisor, and talking with other members of their departments (both formally and informally-- An awful lot of very good work gets done over lunch), and going to conferences where others talk about their work.
And of course doing their own original research. This looks different for a mathematician than for an engineer, of course, but either way, it’s the most important part of of the PhD. When someone gets a PhD, that’s an acknowledgement that they know something about that subject that they can’t have learned from a textbook or a journal article or a lecture or whatever, because it’s something that nobody else in the world ever knew.
There are various “graduate-level” classes for beginning students. And any decent university will have various research seminars and colloquia, of course. Also, at some point pretty soon you will need someone from the faculty as a supervisor. Your advisor may, for example, have you work through a textbook/monographs/published papers/whatever as an individualized study to learn some particular topic; that happens. You will need to dig through a lot of papers, anyway, to familiarize yourself with the state fo the art as well as the history of whatever problem you are working on. You are absolutely required to do some original research for a dissertation, but it’s not lab work like in e.g. biology. You know the joke about pencils, papers, and wastebaskets…
There are quite a few graduate-level mathematics textbooks. For example, Springer publishes a whole series.
The details vary quite a bit from school to school. When I was a grad student, too many decades ago, the first year was all coursework, some of which involved textbooks, all of which involved lectures. (If you took careful notes, that amounted to a textbook in itself.) At the end of the first year you took a series of oral exams, which served as qualifying exams to continue. If you made it that far, you got a Masters degree; if you did sufficiently well on the qualifying exams you were invited to continue toward your PhD.
Where I was the next step involved delving deeply into two topics that you would be tested on, again orally, in the “topic exams.” By this time you had an advisor and the topics were worked out in conjunction with them, with the idea that those topics would be useful background for your dissertation later. Materials used could be textbooks or journal articles.
From that point on, the real work centered on the problem you were going to work on for your dissertation. Likely you needed to dig deeper into the literature to get to the frontier where you would be working. That literature could again involve textbooks, but certainly would involve research monographs, journal articles, and preprints for the most up-to-date material.
Yeah, echoing @Topologist – In most cases in most STEM fields, you haven’t done all the book learnin’ there is to do in your field by the time you enter graduate school. Things are handled differently in different countries, but in the U.S. it is common for STEM graduate students to spend some of their first year, and possibly a lesser amount of their second year, taking classes, and many of those will still be textbook-driven classes.
This doesn’t conflict with your engineering friend’s statement that “most of his doctoral work” was on the research side. That will be true across the board. It just varies field to field and school to school how much up-front course work is still required to build suitable background knowledge. There are also many cases where one must take courses slightly away from one’s intended research area to ensure sufficient breadth of knowledge. (Thesis research is very much “depth” over “breadth”, so many STEM programs require a bit of breadth to ensure they’re producing well-rounded PhDs.)
My ex has a PhD in Mathematics. I have never seen her reading a textbook, except for the undergraduate classes she teaches.
For her research, she always had a couple of journal articles open on her computer. I met her about 5 years after she got her PhD but from what I understand, this is how she’d been working from the start, reading articles and discussing them with her advisor/colleagues. That, and conferences, of course.
I wouldn’t say that I never consult textbooks in my area, but when I do so it would usually be to find references for basic facts I want to quote in an article I’m writing.
On the other hand, sometimes my research overlaps another area that I’m not as familiar with and I might then consult a graduate-level textbook to get up to speed or remind myself of things I might have learned long ago and forgotten. For example, my main research area is algebraic topology, but I’ve recently been doing some work that overlaps with algebraic geometry (a field with very different techniques, even though the names are similar), so went back and (re)read some textbooks in that area.
And the borderline between textbooks and peer-reviewed research monographs, like those in Springer Verlag’s Lecture Notes in Mathematics series, is fuzzy. Monographs aren’t necessarily written to be taught from, but they’re often used by grad students in the way they would use a textbook. I had a book published in the LNM series recently and one of the reviews suggested changes to make it more usable to learn from, even though I wouldn’t call it a textbook.
Yeah, I wouldn’t expect tons of active reading of textbooks from courses taken a decade ago.
This part might depend what you or she calls “the start”. Doing research and ultimately writing and defending a dissertation is definitely the core of a PhD program, but there are more general requirements (…in US departments here; other countries have various different approaches). Some combination of required advanced graduate courses, breadth courses, oral exams, written exams, teaching obligations, etc., are the sorts of things you find in STEM PhD degree requirements. Those are front-loaded in the first year or two, sometimes with a residual amount going on into the third year.
In top programs one would start leaning into research ASAP in parallel with all that (ideally from Day 1 if feasible). Most programs have a concept of “candidacy”, and a student “enters PhD candidacy” or “becomes a PhD candidate” after they’ve finished the core requirements and are working 100% on research.
So, if you or she calls PhD candidacy “the start”, then, yeah, it’s all research-driven work at that point. If the start is the start of the PhD program, then there would have been all that other stuff, too, including coursework.
More generally, PhD programs all have their degree requirements listed online, so you can Google “mathematics phd requirements ” and take a look.
While there are certainly graduate-level classes, that are structurally similar to undergraduate classes with lectures and a textbook and so on, I wouldn’t generally refer to someone who’s taking those classes as a “doctoral student”. They might intend to continue through to the PhD, but for now, they’re still master’s students.
That’s not typical. One has to apply to a PhD program from the start. Some of the early requirements may overlap with those of a parallel masters program, and in some cases a masters degree may be awarded to those in the PhD program en route to their PhD. But it’s not the case that one is “simply” a masters student and then later becomes a PhD student. Masters and PhD programs have quite different admissions standards and expectations. Someone who applied to, and matriculated into, a PhD program is definitely a PhD student. It would not be correct to say they are a masters student, even while they are doing non-research-related program requirements in parallel with research.
Right. I only received a Master’s because I asked for one. This amounted to walking into the chemistry department office and checking a box on a form. Most of my classmates did not unless they left early without a PhD. This can be field dependent.
The physics graduate program I completed had only a PhD program. There was no Masters program. The first year was fundamental physics coursework: quantum mechanics, electrodynamics, statistical physics, classical and fluid dynamics. The second year was advanced coursework matching your subfield: particle, condensed matter, nuclear, or quantum fields. Coursework was entirely textbook and lectures.
You’d start research with a professor from the beginning. Some would start the summer before the first year coursework. Details would depending entirely on your professor.
There was a qualifier you’d need to pass to demonstrate you had strong understanding of the fundamentals. The qual was given twice per year and you could take whenever, but most students only took it after completing the first year and usually after the second.
You’d be a PhD student the entire time. If you failed the qualifier twice, you were strongly encouraged to exit the PhD program and take a terminal Masters degree. But there’d be no department support for you if you stayed, unless your professor was willing to cover your salary.
A Masters degree was simply a matter of some paperwork that you and the department filled out, once you’d successfully completed two years of coursework, whether or not you passed the qual.
She didn’t get her PhD from a US University. I don’t remember her telling me about taking any classes during her PhD, but I may be wrong, or perhaps she never told me about them.
However she did take a few courses simultaneously, but because she was interested in them not because she needed them for her research. Actually, they were in a field that is quite remote from it, although it was still “Mathematics”, very broadly speaking.
Basically, she gave exercise classes to undergraduate students, read tons of articles, attended conferences and worked on her research.
My (engineering) PhD program assumed you already had a master’s degree (which was 24 credits of classes and 6 credits for master’s thesis) required 30 credits of classes (10 classes, most had to be graduate level but some could be senior level, I forget the allowable mix) plus 30 credits for research and dissertation. A full-time student could get the master’s in one year and the PhD in two more. That was rare. I got my master’s in three semesters plus a summer, then took 8 years to get my PhD as a part-time student. That was even rarer.
DQEs were when usually you finished the course work but before you started your research.
Bottom line: there was a substantial amount of class work required for my PhD program. Most classes had textbooks.
I learned primarily from lectures in courses. This was supplemented by long conversations with one professor who eventually became my thesis advisor. My only other source of knowledge was a deep reading of one of his papers that I amplified significantly, while he lost interest in the subject and started a new area of research, while I continued in that area for about five years after my PhD when I eventually cracked open the whole topic and then pursued other things.
Nearly all my published work consists in pushing someone else’s work wider or deeper. So I learn a work and extend. The sole exception is the paper referred to in the preceding paragraph which had no antecedent.