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I have just finished my bachelor degree and I will be entering a PhD program in mathematics. On the home page at my schools (and others) I have seen that there are various groups, composed of teachers and students, who study a subfield of mathematics. For example, I have seen "Mathematical Logic" groups, "Mathematical Physics" groups, "Dynamical Systems" groups, etc. What are these groups for? My view of a PhD was like this:

1) You take a couple years of classes and pass your exams.

2) You find a professor and do a thesis.

How are these groups involved here? Are they just for students in the same field to hang out and talk in general about the field?

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Mathematical research groups are, as you seem to have surmised, just groups of faculty members and students (i.e., graduate students, postdocs, tenure track and tenured professors) whose research interests all involve some common subfield of mathematics. Thus a number theory group will be composed of number theorists and perhaps faculty members whose main interest is in some other area of math (e.g., combinatorics) but involves number theory in a substantial way. Continuing with the example of a number theory group, at some schools all of the members of the group will be involved in the same flavor of number theory (e.g., the group might be biased towards arithmetic geometry), while at other schools the group might have people representing many different area of number theory (algebraic, analytic, arithmetic geometry, etc).

So what is the purposes of these groups? Well, they're just as much for the faculty members as they are for students. As you mentioned in your post, students in a research group are likely to hang out and talk math with one another. (You'll also be taking a lot of topics classes with the other students in your research group.) The same goes for faculty members. If you're a professor working on a project it's very nice to have someone who works down the hall from you that you can ask fairly technical questions to. (Or perhaps "stupid" questions whose answer you don't know but feel you should.)

Research groups will also generally have their own weekly seminars. Sometimes these seminars have a lot of talks by the faculty and students at that school, while other times the speakers are primarily visitors from other schools. This gives graduate students the opportunity to see what sort of problems people are working on at the moment and a bit of perspective about the nature of their research area that goes beyond the particular problems that the local professors are studying.

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    At least in places I have worked, research groups also often function as the smallest organisational unit in the academic administrative hierarchy.
    – avid
    Commented Aug 8, 2019 at 4:09
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    @avid This is location-dependent. In Austria and Germany a research group is in fact an administrative unit. In Sweden, for instance, research groups are completely virtual and exist outside the official hierarchy. I am sure other countries have a mixture of these models.
    – xLeitix
    Commented Aug 8, 2019 at 9:15
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    FWIW, what @avid describes was very much not the case in my previous job (Canada) but is increasingly the case in my current job (UK)
    – Yemon Choi
    Commented Aug 9, 2019 at 2:47
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The groups are a bit more serious than what you suggest. Generally they consist of a few professors and their doctoral students. All are interested in some fairly narrow range of questions. Often they are led by a senior professor. They pose research questions to one another (long term research ideas, not simple exercises) and explore approaches. Often the doctoral students get their research problem ideas from the seminars held by these groups.

Perhaps they discuss papers and work in progress. The currency here is ideas, approaches, insight.

It is a good idea to get attached to one of these relatively early on so that you get some sense about the research that is done and how people approach it. Don't wait until you've passed qualifiers to start narrowing your focus to a professor and a potential topic. These "seminar" groups can be valuable for that.

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    Thank you! If I may ask you a few follow up questions: By the second or third year, are (virtually) all graduate students (and their respective advisors) in one of these groups? 2) Does it happen that you want to do your thesis in a subject nobody else is doing, and therefore you do not have a group? 3) Do the students generally share the research they are doing? I am asking this final question because my TA in undergrad would not tell us the specific topic he was doing, he said he didn't want anyone to take his idea.
    – Helix
    Commented Aug 7, 2019 at 18:17
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    Your TA was a bit paranoid. Maybe with reason, but most people share within a group. Your research us usually guided by an advisor who may be in one of these groups or not. A few people can work without advice, but it is rare, and harder. I doubt that all are in a group, but it can be helpful if you share interests.
    – Buffy
    Commented Aug 7, 2019 at 19:51
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Just adding to other answers and comments: it'd be unfortunate in several ways if a PhD student didn't talk to anyone other than their own advisor. For one thing, they'll need a PhD committee of 3 or 4 people who're willing/able to endorse the thesis, and also hear the defense.

Oh! And the letters of recommendation to get a job!!! You'll need at least 3, and after your advisor that leaves 2 more. If you've never talked to any other faculty at your home university (or anywhere else) you'll get lukewarm letters. Truly, you'll want some other faculty to be aware of you and what you're working on!!!

And getting a broader education/exposure in your general subject is a good thing... and perhaps the most convivial for beginners is a seminar run by and given by beginners... "from your group".

Finally: as with many human activities, while one may find reason for it, to be "without a group" is generally dangerous... since one has no support, no encouragement, no validation, no sanity checks...

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  • You need that many recommendation letters for an academic job, not for every job.
    – user111388
    Commented Aug 8, 2019 at 15:38
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Groups are exactly that: clusters of people usually with common interests. They can be important for graduate students as it can allow students to share expertise, try ideas (of proofs if you are in math) and broaden the knowledge base if only because others will have a different background, will have read different books/papers etc.

The same holds for faculty. Some groups will have specialized seminars and share in the cost of hosting such seminars (formally or not) etc.

Basically it’s an easy way of pooling resources to the benefit of all.

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