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This is my personal take on applying to math doctorate programs in the US as an undergraduate student at an US college. For privacy reasons, I want to focus on the less personal aspects of grad school application (i.e. advice, facts, etc.) rather than my personal profile (i.e. my relationship with my letter writers, classes I’ve taken, etc.).
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I recently got a email regarding my Phd application and I thought it might be beneficial to share some of my own experience.
I started writing my Statement of Purpose (SoP) as early as mid August and finished my first draft in late August. Most of my SoP’s are similar beside the focus (AG or NT) depending on whether the school is strong in NT or not. The Personal Statement (PS) is required for some institutions such as University of Michigan; as opposed to my SoP which discusses my academic journey, it describes my involvement in the mathematical community in general. Although my timeline sounds a little too advanced, I prefer to keep my priorities straight, and there is nothing more important that summer beside writing up my SoP. (Translation: I had nothing better to do at the end of that summer). Writing with a blank slate is quite terrifying, but I’m sure that math phd applicants have already written a handful of REU PS’s so that they have something to start with. Gradually over the semester, I filled in the applications forms when I was bored and it didn’t take up a lot of my personal time. It was some mental load though, even though I submitted most applications well ahead of time. I wouldn’t recommend taking too many credits during that semester.
I remember reading this on academia stackexchange one day: this committee member complaining about the point of SoP’s because they can make the whole decision based on the CV and transcript. One of the SE users replied that SoP’s are really not that useful, and are mainly required because it gives the students something to work for during application season. Regretfully, this is probably true. I wouldn’t expect anyone to read my SoP for over 30 seconds even if someone do read it. However, I do enjoy writing up about my academic journey, and it gives me a sense of achievement of how far I’ve come. The general rule for me is to be terse about “why I love math”, and instead write about what I’ve learned (mathematically or philosophically); being an avid reader certainly helped with the process.
Letters are much more important to your application as compared to the SoP. One thing (that I won’t go into detail) is that it’s important to have one or two plan B candidates for your letter writers, because they might not write a letter for you for various reasons. I followed Vakil’s advice on letters: send everything they’ll need ahead of time, including all my application materials, specific information that might help with the letter, and a list of schools and deadlines. I applied to plenty of schools and some students are probably afraid of asking the same three professors for that many schools, but it’s fine in general. After all, everyone does that! Also, make sure that you remind the letter writers to submit the letters about a week before the deadline; letter writers are busy and it’s common for them to forget some deadlines when they are writing LOTS of letters for people of all levels.
I’m generally an anxious person, but I managed to keep my cool through the application cycle because I made a conscious effort to avoid comparing myself with other people’s profile. I never looked at forums like mathGRE or gradcafe, knowing that whether or not my profile is better than theirs won’t make me feel any better. I am confident that I have a good profile given where I started, and I’m also well-aware that the application results are a combination of various factors outside of my control. This mindset has helped me to navigate the stressful period of writing and submitting applications without too much self-doubt.
Grad schools will begin releasing their results starting as early as early to mid January. Again, I avoided discussing my application results with my peers who are applying in the same cycle. One thing that I was particularly looking forward to as a junior is that I want to get into lots of grad schools, come in for their open house and travel for free. I did get into many grad schools in the end (given that I applied to lots of them), but this dream didn’t come true for a variety of reasons. I couldn’t drive, which means it’s hard to get to and from lots of colleges that are hours of driving distance from the airport or bus or train stations. Some open house events have inconvenient times (i.e. right after my in person midterm) which means I can’t come in on time. Some schools have strict upper limits for reimbursement and it’s hard to afford plane tickets or hotels on that stipend. I ended up only travelling to two such events. Whether or not the open house is in person, I strongly recommend travelling to your top choices after getting admitted because it’s important to see if you want to live there for at least five years. It’s also beneficial to talk to your potential advisors — at this stage, the faculty at the department will probably happily welcome you because they want you to be there. There is also no shame in visiting grad schools knowing that you won’t commit there; it’s always good to know people from different math departments.
Personally speaking, the post-app season was a harder time because I need to deal with more realistic factors when making a decision. Some grad schools are good in math but offer a low stipend that would make it hard to live there comfortably; have an undesirable location; or has a heavy teaching load at the beginning of grad school. My observation is that many public schools seem to have larger apartments but offer lower stipend and require a heavier teaching load such as being an instructor; whereas private schools have higher stipends and only require TAing. Another thing that my undergraduate advisor reminded me of is that larger apartments usually have the same number of people actively working in the field I’m interested in compared to smaller apartments. Therefore, private schools (which generally have smaller apartments) are not necessarily worse academically. I’m very fortunate to be admitted by Northwestern, which is suitable for me academically and otherwise; but in hindsight, I would gravitate towards applying to more private grad schools.
My undergraduate advisor offered me many great application advice including the one about departments, but the most useful is probably about spamming the departments during the waiting period. They told me that unlike the postdoc application (which relies heavily on networking), the grad school application is more rigid and contacting specific people in the department is not so helpful. However, it’s helpful when you are on the waitlist because this will demonstrate that you have a strong interest in getting into that department, and it can potentially result in yourself moving up the list. My advisor also told me that getting on a waitlist is not a death sentence and it’s worth trying. I’d like to think this advice helped me get into Northwestern, because someone at the department immediately replied to my love letter, and they called me that evening telling me that I’m in before the official offer.
I am very privileged to have got into one of my top choices for grad school, but I think I would have committed to another doctorate program instead of re-applying the next cycle. I generally advise against applying only to the top choices because it would take at least two years for you to have some meaningful change in your profile, suppose you want to keep applying for a math doctorate. While I don’t think there is a penalty for reapplying, it certainly doesn’t help that there are few research opportunities for masters students in maths or students on gap years. I also think the less-preferred options are not necessarily bad; in fact, I am very grateful that I got rejected by Northwestern engineering (my then dream school) as a transfer student because that would have led me onto a very different path. Finally, if you want to move out of maths unless you get into the top grad schools, maybe you shouldn’t consider applying there anyway — there’s a good reason that you don’t want to do math, and getting into a top grad school probably won’t change that.
While I never suffered from anxiety throughout the application cycle, I felt a sense of loss with a pang of impostor syndrome for the next eight months before starting grad school. I was out of place for both undergraduate opportunities like REUs or graduate opportunities like summer schools or conferences; and I was not learning anything particularly interesting to me in math for quite sometime. My friends in the honor math program are drifting away to grad schools outside of math or industry, and the ones admitted to a doctorate program in math seem to be busy with their own plans. As a result, I committed to some experiences outside of maths besides preparing for the prelim exams that summer, but none of these experiences genuinely excited me. There was a time when I, gaslighted externally and confused internally, thought that I had lost my ability to do maths and it was a bad decision to get a doctorate. After all, it was during the post-app season of transfer application when I decided that I should quit engineering and do maths three years ago. (Ironically, I was rejected from all schools of engineering at the time. )
In hindsight, this feeling was an avalanche of emotional responses to the ending college life and the stressful environment in which I spent my summer. While I still struggled with the non-math commitment that extended into the first quarter, I am now regaining my faith as I’m doing some interesting guided learning, connecting with the math community at Northwestern professionally and personally, and planning my travels for mathematical events. I am genuinely excited by the fulfilment and challenges in maths and grad school, and I hope I retain this faith in times of uncertainly throughout my grad student career.