Paraphrased and considerably shortened excerpts from an email correspondence (between me and a friend) about PhDs, academia, studying, expectations, and more. The excerpts here focus mostly on our viewpoints on attending mathematical talks (April 2024). Some good parts are highlighted.
1: I haven’t managed to get into the habit of going to talks; you often seem to learn very little from the talks compared to the time you spend attending them. But I agree that it sometimes is nicer to hear someone talk about their research than to just read their paper.
D: Indeed, meeting researchers in person seems important - I was told by a friend that, in their field, there aren't many accessible neatly written papers, so they'd been advised to instead learn from people e.g. at conferences by asking about their papers.
I would still say that talks are useful; have you checked out Ravi Vakil's "3 things"-rule for ensuring you get something out of talks? I haven't really applied the rule myself; I've just noticed that you often take away around 3 things from a talk, and these things will accumulate. The good thing about talks is that you can hear what researchers find interesting and ask them questions! I find it quite intimidating each time I ask a question, but that means you gain from practicing…
Going to talks will probably later be a way for you to get in touch with other researchers, ask them about works relating to your own, et cetera.
And you will surely go to a lot of conferences as well. Doing math is not all about what you discover in your own time, but you will gain both from discussing and getting contacts to have more people to work and interchange ideas with. You will have the chance to make people interested in your ideas as well.
Even though individual talks I've been to haven't taught me much, I can still feel an immense gain from all of them collected together; it has made me find interest in learning particular areas and given me a feel for what the hot topics are. And I've also learned a bit, also about how you give good talks. It is also a nice environment, you get to chat a bit with people if you want.
1: Your comments about talks made me rethink whether I should start going to more talks. However, there are more things I would prefer to do in an hour than impatiently sit in a talk and barely understand it; many positive trends in my life come from prioritizing other things than math in my free time. I consider myself a physicist and not a mathematician, and when I learn math or physics, I prefer just learning the basics and fundamentals really, really well and taking things at a slower pace.
You mentioned setting a timer for 40 min, which reminds me of the so-called “Pomodoro Technique” which I’ve been taking up again for breaking up work into chunks. During the peak of my productivity, I didn’t really need it, but it certainly helps with focus and for making sure you take regular breaks. I am, however, surprised when some people say that they work for most of the day, since it sometimes can be difficult for me to get more than 4-6 hours of useful productive work out of a day.
D: I rather set aside time than set timers. Yes, it is helpful to not work without breaks for hours; I usually resolve this just by doing different tasks each hour and have a few minutes break in between full/half hours, as well as making sure I spend at least one hour outside per day (ideally I would like to be outside much longer, but the weather sometimes makes it a nonsuitable study environment).
I usually also find so many things I need/want to learn that I really have to view my timetable as a puzzle and try to fit everything nicely. You can often gain much productivity from making a clear schedule and following it.
Many people might be more productive with shorter workdays, but to me this seems more applicable when you have less tasks to focus on and that you don't know precisely what to do (unless you work so much that you get tired, but it is often possible to extend your productivity e.g. by taking regular runs).
Of course, learning preferences are personal. Sounds good that you know how to learn best, i.e. learn the groundwork well. I think the best way of learning is through your own interest and exploration, whatever that might be. But it might still be relevant to see what other people work on/find interesting etc. I guess to - prevent you from spending your lifetime on something that has already been solved (although I don't see anything wrong with this, actually, as long as you discover math on your own!) - help you find interesting directions (again you might be able to do this completely on your own, eventually) but also to find collaborators & to talk to others about your topic (you can't do this on your own, for certain!)
It is useful to be updated in your field and to keep contacts (not just for "making progress" but also because it is nice to have a social learning environment).
There are surely talks that are more valuable than others. Maybe your research group will host a regular seminar series?
You must certainly have been to some talk in your lifetime that you found interesting or learned something from? Even if it is just an "idea" or motivation to learn a specific topic?
Is it really that different being a "physicist" or "mathematician"? (Technically, algebraic geometers are doing string theory sometimes...) Is the "reality aspect" that important? There are myriads of interesting geometrical phenomena that you can think of without them being real-world-based (and a lot of physics is also hypothetical & hence abstract?)
As for patience, I would not say that I am impatient but rather that I find too many things I want to do so that I seldom sit down and focus on one task entirely (with exceptions for my research projects), which sometimes leads to "many things done briefly" rather than "one thing done well", or equally likely, "a way too overloaded schedule and many things done well".
I agree that you shouldn't let your work define you (because we have so much potential to "be" in so many ways at the same time! And life has many things to teach you.)
D: What did you think of Donaldson's talk?
1: In order to work for 10 hours in a day, it feels like I would need to break the laws of physics! I don’t understand how this can be possible… I don’t particularly like the idea of making schedules either, you just constantly feel busy - I’d much prefer to feel free! I’ve also started to think of what to do with my time.
This “discovering things yourself” is something that seems beyond reach to me. I would be much better at digesting and explaining other peoples’ work. Some people claim that they have predicted the theorems coming up in lectures before having seen them; I don’t understand how they can do it.
You really seem to want to convince me to come to talks, and surprisingly I enjoyed Donaldson’s talk! Maybe I might try going to more talks in the future, if I suddenly have excess time. It can, however, be a bit stressful to try to follow a talk that goes too fast.
Yes, there is a huge difference being a physicist and a mathematician (string theory is maths)! Having the label “physicist” is important to me; physics has always felt like a welcoming environment where I just get to have fun learning about cool things.
And the reality aspect is very important to me too, I think. With mathematics the definitions often feel arbitrary and unmotivated, even if I can understand them. After all there are infinitely many different things you could define and study. So I have to just have faith that these ones are interesting and worthwhile pursuing - but that requires me needing to trust mathematicians... I think physics to me is a label that should be respected, and I get uncomfortable when people attach it to things that are too speculative and not grounded in reality. Everyone should be able to enjoy and see physics, and I'd much rather it not be locked behind abstract math, especially if you can't even test it feasibly!
I do not like the use of the word “potential”; it reduces you to what you can do rather than who you are as a person! I do not want to be overestimated either.
D: When speaking about "fundamental research" or "own discoveries" (which both sound quite grand and to pursue them is easier said than done) I do not refer to big, important results, but rather the process of asking and investigating stupid questions that might only interest yourself at the moment, and which may arise e.g. in papers you read, rather than being discouraged by the fact that "someone else would have found that stupid, trivial" etc. By doing this I would not hope to discover anything major, just satisfy my own curiosity.
Again, going to talks might - whilst you're sitting & trying to make sense of formulas you have absolutely no clue what they mean, and your brain tries to oversimplify things or find the idea; the more lost you are the more you come up with (completely stupid) ideas why the things might be meaningful - make you realize what aspects you would like to examine to make it meaningful to you rather than necessarily following the "given path". Maybe you'll just end up realizing that the given path is meaningful, and in this way you learn both the result and why it's meaningful rather than blindly following the given methods. Or, alternatively, you can overload yourself with theory, definitions and facts and hope that things will fall into place, but it might be an unpleasant way of learning.
I think you should not be worried about "not coming up with new things"; (PhD-students seem quite stressed about this though...) as long as you're curious about what you're doing, and you let your curiosity guide you, I think that is good on its own. And always at the end of your PhD (say 3rd year) you'll have found something (because literally everyone does) so you won't be empty-handed. It is probably better to enjoy your PhD rather than being stressed about publications (again, easier said than done?). Then it just so happens that the grand results come through the "exploration"-approach, by chance maybe. The more stupid questions you ask, the greater the chance that something will be of value, I guess. But again, it is not the "groundbreaking results" that are the only things to strive for, neither is it worth striving to "publish something and get some results just to have published something", or pursue a task just because someone else told you or claimed it was important if you're not curious about it yourself.
It might be this kind of "publication stress" that makes math overloaded with things which are seemingly random and not clear why they are interesting. On the other hand, I think that many of these things might be interesting, but maybe there hasn't been effort put into showing why, or maybe they are just interesting to one person. I think a good thing with math is that there are no limitations; only human creativity can set boundaries for what there is to discover. And you don't need to have reality as a motivation for claiming that something is interesting. You can draw pictures and use a particular method or explore a result just because you feel for it, like artwork.
And in the end, through small steps, we are expanding our knowledge about the unknown, maybe with the ultimate goal of, which is certainly not attainable within the time scope of human existence, understanding our universe, our limitations/possibilities, meaning of life etc etc
About the math community, I quote Grothendieck: (about peoples conviction that they themselves know nothing and everyone else does) "It is this doubt, this intimate unspoken conviction, which drives them all to constantly surpass themselves during the accumulation of honors & works, & to project onto others this contempt for themselves which secretly gnaws at them by the accumulation of "proofs" of their superiority over others".
There are many problems with the math community, and I'm glad you find the physics community welcoming. I'm trying to become aware of these problems and hopefully be able to improve them at some point, and hopefully people won't flee into physics but actually stay and help change the situation... (Although physics is of course very interesting, so I do not blame anyone who stays in this field...)
About "potential": Indeed, it is quite annoying to be judged by prestations etc, which also can lead to a performance-based self-esteem. It often feels like people estimate your ability (either overestimate or underestimate), and it is not only the "over- or underestimations" being annoying but the fact that they make estimations at all. The thing is, that if people seem to think you're bad at something then that gives you less motivation which means you actually perform worse.
Ideally, we would have a math environment where everyone is encouraged, regardless of judgements of "ability", because ability depends a lot on motivation (and the better the environment, the more people will join and the better the progress). When people overestimate you, that can also lead to overcompensation with studying too much just to live up to the expectations.
For me, the word "potential" has a slightly different meaning; I see it as something most people have, and is something that should be taken care of, making people eager to use it. As for myself, I find it comforting, when things don't go so well, or when "everyone else seem to do better", to think that I have a "potential" that will make it go well in the end anyway, not because of pure ability, but rather a deep motivation and curiosity knowing that I won't spend much time on "being tired of math" which unfortunately seems quite common among students for some reason.
About labels: I probably don't use labels too much since I cannot put myself into any category so easily, I guess with the only exception being "mathematician". I am even unsure how well the label "Swedish" even suits me (given the time I've spent abroad…). Even with the label "mathematician", I am sure there are many ways this can be interpreted which would describe me less well. Oh, actually, "algebraic geometer" suits me quite well.
Maybe you should give a talk at some point about improving the math community? I am sure it would be appreciated.