Hi,
I’m a math major planning on starting a PhD in mathematics next year. I have good grades but am worried that I won’t be prepared for the jump in difficulty, since I never tried much in my courses and as a result have poor study skills. Is there anything I can do to improve this?
Thank you,
syntacticalbeing
Read Cal Newport’s writings on Deep Work, or just buy the book.
A Math PhD requires a research skill set, and you should like Math research. Make sure you like research before going down this path. Otherwise it can be unpleasant. The big difference between undergrad math and PhD math is that in undergrad, when someone gives you a problem, you know that it has a solution. In PhD, both you and your advisor do not know whether a particular problem has a solution. You are finding new results. It could be exhilarating. It could also be incredibly frustrating if you don’t see any progress for months or longer.
I do like research. I spend a great deal of time working alone on topics that interest me, but it’s difficult to make any progress.
It’s still difficult in grad school. The second most important thing* about your PhD program is your supervisor: s/he will make all the difference in managing the difficulty and the process. Math PhDs can be very unstructured: more than study skills, you need the self-discipline to keep working at it, month after month, year after year, in the hope that you will get there.
Soon-to-be SIL is in year 5 of his Math PhD in a T10 program, has been working steadily and reasonably fruitfully right the way along- but has only just now found the nut of what will become his doctoral thesis over the next year. The hardest part for him has been how relatively solitary and unstructured the process has been (he is fairly extroverted, and of course Covid didn’t help).
*first, obviously, is liking your subject & research enough to spend 6 years doing it!
I have no problem in terms of interests. The complete lack of peers with similar interests made it far more difficult for me to learn than it should have been, and yet I’ve been able to make some progress. This has been going on in math specifically for more than three years, and in other subjects for my entire time in grade school and college. I expect that aspect will be challenging in grad school, but not as bad as what I’ve had to deal with before.
Just to be clear, doing research to prepare for grad school means working under a professor doing formal research. You’ll likely need this experience to help you get into grad school, not to mention the connections you’ll make will help you be able to get strong letters of recommendation. You made a comment about spending time alone working on topics which made we think you weren’t doing research with a professor.
I have been doing research formally as well, however my interests don’t align that well with the faculty, so I’ve been spending lots of time outside of it self studying.
That’s great.
I have a slightly different take on this. Doing a math PhD was the least “work” I’ve ever done. But it was the most “thinking” I’ve ever done. Do you dream math? Have ideas come to you in the shower that you need to write down? A friend in college (who is now a fairly famous math professor) told me you need 2 hours of inspiration to get a great math PhD. Mine wasn’t that good (maybe 1 hour of inspiration 
) but the key results were derived in the bath (it’s good to have both fingers and toes available for proofs by induction) and in bed while at a rowing regatta. It’s just a question of having things in the back of your mind and waiting to (hopefully) figure it out. To paraphrase a quote that I see used a lot about current geopolitical events “there are years when nothing happens, and days when years happen”.
So if you “spend a great deal of time working alone on topics that interest me, but it’s difficult to make any progress” I’d want to know whether you ever find a solution just coming into your head, and whether you’ve spent much time on/had success in doing the sorts of activities where that is required (eg Putnam exam questions). It doesn’t just have to be in math, even today that’s how I figure out problems at work that have nothing to do with math.
I do agree on the importance of finding a good supervisor (mine was mostly terrible). I hope you are identifying people you’d like to work with based on your area of focus. It will be much harder if you’ve just done a range of different things or if faculty at your current school don’t cover the topic you want to specialize in and so you can’t get a recommendation from a professor that your potential supervisor would recognize.
that is a really apt description of life for the Math PhD students I know!
I do manage to get moments of clarity when working alone. I’ve been having some trouble because the topics I’m interested in are pretty far removed from anything the faculty work on, and the only books available are very dense/intended for experts. While at college I’ve done a range of things and haven’t been able to get as far using classes and official research since there just aren’t enough peers to do so. I finish assignments with minimal effort and spend lots of time on basic things when doing official research since less background is generally assumed for those projects. I’ve tried the Putnam but am bad at test taking and haven’t really prepared for any of the exams (they were just for fun). I originally wanted to major in physics but chose to switch to math since none of the physics faculty do work that I’m interested in.
It feels like you need to do something this summer with a professor in your area of interest at another university if you are going to establish a track record that would allow you to progress to a good PhD program after graduation. It doesn’t seem that you have the support in your current college that will enable you to have strong recommendations. Math is very prestige sensitive so a PhD at a low ranking university can make it hard to have an academic career. Consider doing a prestigious masters (Cambridge Part III being the classic example) if you can’t get into a good program.
What exactly would be a “good program” for a PhD? Rank ~50? Rank ~20?? I have a perfect GPA, have taken multiple graduate level courses in adjacent fields of interest, and have solo authored a peer reviewed paper before starting college, but I highly doubt I could get into any prestigious program, let alone anything like Cambridge Part III. I barely even managed to get into a state school for my B.S. I have no problem with the academics and am generally viewed favorably in person, but admissions has gotten so competitive that I don’t think it’s enough.
Top 20 unless there is a well-known professor elsewhere in your area of interest. Why don’t you start reaching out now to professors who are in your area of interest and ask what they are looking for? Ideally you should have already been asking if they have opportunities to work with them this summer, especially if research in your area is limited at your current college.
Does your college participate in the Churchill scholarship competition (https://churchillscholarship.org/)? A lot of those students do Part III.
I have applied to graduate schools this academic year and have gotten into a program with someone fairly well-known in one of my fields of interest, but it’s not top 20 (more like ~50 or so). I’ve never even heard of the Churchill scholarship. My college has advertised scholarships, but I don’t qualify for most of them since I’m not any type of URM. I was recommended for the Goldwater scholarship, but it was too late since I was a senior at that point (graduating in 3 years instead of the usual 4). I’m mainly concerned that I won’t be able to do any work in my field of interest due to program reputation, even though I may do well in terms of the actual research.