How does working with other people in math actually look like?

Question

I'm about to finish an undergraduate degree in mathematics, and I want to do research in mathematics, so naturally I'm looking into postgraduate study options. I'm quite stressed about this, and I was thinking about what I want in an institution to continue my studies, and I came to the conclusion that all I want is a sufficient stipend and freedom from major hurdles. The issue is, I look around me and see people having collaborations and talking to each other, and none of that is happening to me. My experience with talking to people about math is full of experiences like:

• Professor presents a concept in class. I ask a question about it, and he says he doesn't know and will look it up and tell me. This never happens. I eventually research the topic and manage to figure out on my own, and it's not hard. This one has happened 3 or 4 times.

• I'm taking a graduate level class and the evaluation is an open ended assignment, and after completing it the professor tells me mine was the best in class, and tells me she wants to talk to me about it later. This never happens.

• I think of an elementary problem, and an idea to solve it. I think the idea looks promising, and mention it to a professor who does research in the area. He tells me it's interesting and encourages me to look into open problems, but doesn't give me any sources to read or mention it ever again.

• I have an undergraduate project to study a topic under the supervision of a professor. I'm supposed to present what I've studied to the supervisor once a week, and by the time I've covered the basics and advanced on to things that he doesn't know, the advisor looks clearly disinterested and doesn't notice when I make errors.

• I have another project like above with another supervisor. He is more interested than the previous one, but often insults me saying things like "I'm happy you're finally going to learn some mathematics" and telling me his other students are better than me.

• A classmate and I decide to study together, usually before a test. We ask each other to solve some exercises we weren't able to do by ourselves, and I'm able to solve some of theirs, but they aren't able to solve any of mine. They aren't ever interested in discussing math in other contexts.

• I ask a question in class, usually about the intuition behind a certain object. The professor gives a curt and unhelpful response. I eventually manage to figure it out reading a book or watching a lecture on youtube. I notice that the professor obviously knew how to give a good answer but chose not to for some reason.

• I attend a conference in a prestigious university far away with lots of famous mathematicians. I can't understand any of the talks, and while I manage to have good conversations with some of the people there, none of them are actually about math.

To me, mathematics has always been a thing that has happened between me and a book or a computer. I have an article published with a novel proof of a classical result, and when that happened I was incredibly happy, and I want to feel that more times. I really do love math, but to me academia is little more than an someone who is willing to pay me to do research. This makes me concerned. I feel like other people are getting opportunities to collaborate with each other that I'm not getting for some reason that I can't understand. It isn't any of the obvious reasons you might imagine, my grades are "excellent" according to everyone who has opined, I got grants that only a small minority of students get, and my professors and classmates often praise me. Even that professor who said I didn't know any mathematics said in a recommendation letter (which I had to see for bureaucratic reasons) that I was "the best student [he] met in [his] carreer".

The immediate decision I have to take is whether I want to go to the university that will give me a PhD for the least amount of effort, or if I want to take more conventional considerations into account. For example, I'm considering choosing a PhD advisor who I know will be absent all of the time. But deeper than that, I feel like I need to understand what is going wrong, and whether I can (or want to) fix it.

Answer 1

First thing to keep in mind is that it is very rare for an undergraduate student to have the inclination and aptitude to engage in mathematical research. As such, it is unsurprising that you were unsuccessful in finding people to talk math research with amongst your peers. This will also colour the expectations of faculty members you are engaging with.

As Zach H already pointed out in the comments, your first three examples should be read as invitations to you to arrange further contact and talk more. Following up is on you. This is particularly true for the second example.

A PhD advisor is committing to much more than an undergraduate project advisor, and typically has much more freedom in this choice. I would expect the typical PhD advisor to be far more interested in their students projects than the typical undergraduate advisor.

In my personal experience, most mathematicians are very kind people. I advise staying away from toxic personalities, better mentors are out there. (Details of "How to find a good mentor/PhD supervisor" are another question that depends on the system and has been asked before).

It takes quite a while to get to the point where you understand conference talks.

Answer 2

It's hard to respond to your experiences point by point without much more information than you can provide, so I will offer an answer in general terms.

You seem to be a good mathematics student, capable of independent work. You should do that work as part of the mathematics community, not with an advisor who will leave you alone, not at an institution where you can get your degree with the least amount of effort.

I would probably recommend that you apply at a good school (look for guidance from professors you know). In your first year or two you will mostly take courses. Pay attention to which professors work in areas that interest you, and talk to more senior graduate students about how they work with their advisors.

PS As for not understanding talks at a conference at a prestigious place - join the club. It's quite common. The talks are often highly technical, designed to communicate with active researchers in the field.