What more should I do to increase admission probability to a top math graduate program?

Question

Everyone learned in high school that admissions to top colleges for undergrad is not a meritocracy because there are simply too many qualified candidates and not enough spots. After getting rejected from 9 top colleges and having to go to a state school, I knew that that at least would change for graduate school because achievements in college are easier to compare, harder to fake, and represent real effort. But now I have to question this belief given how acceptance rates are below 10% and taking tough courses, having a high GPA, and having great recommendations doesn't set you apart from the average applicant. So I wish to know what I could do that I'm not already doing or planning to do. To that end, here is a list of some things I've done:

• Courses taken by the end of this semester: MATH 340, 341, 403, 410, 411, 432 on this list, and STAT600, AMSC466. A+ in all courses so far.
• Established a good reputation with one professor after excelling in a seminar and a few math competitions, these being organized/proctored by him. Still need one more reputation like this to have 2 strong letters.
• Joined a directed reading program, currently reading through Graph Theory by Diestel.
• Published papers in high school journal and on arXiv.
• 4.0 GPA, more than half of credits towards degree done.
• By the end of this semester, will have satisfied all requirements outside math except for one writing course, and will have satisfied all criteria about specific math courses or topics. Thus, I have time to take plenty of graduate courses and the freedom to choose.

Here is what I'm planning to do:

• Summer 2021 REU. Didn't get into any REUs this year, probably because I only applied to several programs and had just one recommendation letter.
• Do even better in math competitions, simply for the fun of it.
• Take math subject GRE sometime in 2021 just to get it over with. I looked at a few practice tests and the questions look easy even once the time constraint is taken into account. However, I will still officially time myself on a few tests just to not let my guard down.
• Take more graduate courses, graduate with math department honors, and maintain a high GPA.
• Ask professors interesting questions outside of lectures so that they remember me. An instructor for a course I did very well in last fall told me he wouldn't be able to provide a good letter because interaction outside class was minimal (I never went to office hours or sent an email). This strategy should make me more than just a student who showed up to lectures and got good grades.

Is there anything significant I'm missing that would make me stand out among the applicants to top PhD programs? I heard that students from lower ranked colleges are expected to have higher statistics, but my academics with regards to grades and course offerings already seem to be maxed out, leaving only the option of making muself look better outside of courses. Surely the answer isn't to just continue doing what I was planning on doing given how competitive admissions are in the 21st century. These lists aren't complete, so a correction will be made if someone I already had in mind is mentioned.

A note about time: I should be on track to graduate spring 2022. If I don't get into a good math program for fall 2022, is taking an extra semester or 4th year and then reapplying for spring/fall 2023 a good idea?

Answer 1

I have some very serious doubts about your first few sentences but ignoring those, I can say it seems like you're in really good shape. To add to your list (as a math graduate student myself) really focus on your GRE. A good GRE score will not help you that much but a bad or mediocre one can certainly hurt you. This is something you definitely can study for and you should aim for as high as possible since you have time to study.

Also, you might want to consider the area of research you're showing interest in. I have no idea if what I'm about to say is actually true but I think PhD positions are awarded based on funding availability and if you are interested in a "popular" area you might have an easier time getting in. In this case probably funding for applicable fields is more than funding for purely abstract fields.

If you can publish more papers as an undergrad that will definitely make you stand out because it shows you can do original research and are less likely to drop out of grad school. Getting involved with projects with your current professors is also a good way to get good letters. The vast majority of graduate students I know did not have any publications when they were undergrads. Having publications shows the school is not taking a big risk on you since many students with good grades are not very good researchers.

I also just want to say, that it seems like you're focusing a little too much on the ranking of the school. You should also consider what kind of math you want to do. I spent a semester as an exchange student at one of the top math programs in the world although my own program is not nearly as highly ranked. The students there were really stressed out, competitive rather than collaborative, and not very happy. Burnout is not uncommon there. This made me appreciate my department a lot. If you are interested in an academic career, you should focus on preparing for a long and fruitful publishing history which means doing deep, interesting, influential research that will land you good jobs in the future. That outcome has a lot to do with a) having an advisor who supports you, guides you, and opens career doors for you and b) having time to dedicate to research. In addition to the ranking of the school also consider whether there are professors you want to work with there and the teaching load you will be expected to cover. You may want to contact a professor's graduate students and ask what their experience has been. Most of the time we will be happy to give you honest insight. Also if you are spending all your time running calc workshops, you can't do much research.

Along those lines, go to as many seminars as you can in your undergraduate department. This will give you a better idea about modern research directions and you will be able to write a more coherent statement of purpose.

Consider applying to an NSF grant for early career grad students if you are a US citizen, or find similar grants in your country. https://www.nsf.gov/funding/pgm_summ.jsp?pims_id=6201

Lastly as for your comment about taking more time instead of graduating early: I don't see any reason you should rush getting out of your undergraduate institution if you didn't get into the perfect grad school so yeah, I think it's a good idea to take an extra year and improve your application while taking more classes.

Edit to address what you can get out of seminars as an undergrad: This obviously depends a lot on who you are and what you are studying. First of all, even experts often don't understand everything or even most things in a seminar talk because mathematicians are generally very specialized. Here are some methods I employed as an undergrad and still use during talks:

1. When you go to a talk, aim for an understanding of the point rather than the details.
2. During the talk, make a list of words you don't know and look them up after the talk.
3. I found it very enlightening to read the oldest papers in English I can find about the subject. For example, I just went to the Wikipedia page about the Eichler-Shimura isomorphism which I have never heard about before, and downloaded the earliest paper in English I could find on the internet. It was this one: https://www.ams.org/journals/tran/1961-100-01/S0002-9947-1961-0140126-3/S0002-9947-1961-0140126-3.pdf Even though this is nowhere close to my field, the paper actually did an okay job of describing the motivation of the field. Humans are social creatures and this is what makes math difficult for many. It really helps to know the history and the stories behind any topic before jumping in. If you can research some of that history then the problems in modern math will have a lot more pull. For example, the subject you mentioned seems to have something to do with isometries of the hyperbolic plane. This is a great place to start for an undergrad. If you try really hard to understand each talk I guarantee you will come away learning a few things and after a year of talks, you will have some idea of modern research problems. This will help you begin to develop your overall interests as a researcher.
4. Also don't expect yourself to understand too much. You want to know is this a classification problem? Is it a generalization of something? What are some applications? The goal is to just learn some new things each time. You're not going to be able to stuff all of algebraic geometry into your head before hand but you can learn the basic motivation and see what piques your interest.