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So, this is a question that I do not know much about. I might get called on not having done a ton of research on graduate school, but I am really curious about this. I am an undergraduate student of mathematics and I am certainly planning on pursuing a graduate degree. I have looked at the top schools of math and they are certainly what one would expect: MIT, Harvard, Princeton, Yale, etc.

My issue is this: Academically, I was a late bloomer. I did not take high school seriously because it was annoying and elementary. Where I grew up, my school did not offer much in terms of AP classes or other opportunities. So, the only thing I excelled at naturally was math classes. Of course, they were extremely easy and required almost no effort on my part, which is what made me leery in terms of taking college level classes. I figured that I was good at high school classes because they are simple, but college level will be a different story.

I was wrong. In university, the gap was wider. Once I became adjusted, I actually enjoyed mathematics because it was so much more abstract and complex. This caused me to do rather large amounts of independent research in graduate level topics (topology, analytic number theory, analysis, etc). My only issue is that I am not in a competitive environment, really. The students are not top-notch, and don't share the same drive that I have. This is why I feel like being at the top of the class is not really an accomplishment, since I am not doing anything that any undergraduate at MIT, Harvard, Yale, etc, couldn't do.

My question that I pose here is this:

How good do I really need to be?

I have a passion for mathematics and for the most part, I seem to be talented at it. In terms of GPA, I have maintained a 4.0 with little effort just because I typically have already covered the material in the class on my own. However, there is a voice in the back of my head that keeps telling me that I only stand out because I am at a school that is not top tier, ivy league.

What does a graduate student at a top tier school look like? In my mind, I see someone that has been publishing papers since their teens, and was at the level I'm at now coming out of high school (I am a rising senior at the moment). Is this true? Am I being realistic? I am concerned because I would love to apply to some of these schools (and to get in is my dream!) but I honestly don't even know what level I am at in comparison to the rest of their applicants. I would hate to get shot down almost instantly because I am a joke in their eyes.

Sorry if this was a bit of a ramble. The question boils down to the title really, the rest is just elaboration for those that are interested. Thanks!

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    ... ivy league. — Don't worry about which athletic conference the school's sports teams play in. – Mad Jack Apr 22 '15 at 23:42
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    Is the question how good at math I have to be, or how good does my application need to look? Either way, it is unlikely to be appropriate here. Also what is a "rather large amount of independent research in graduate level topics "? Does that mean you have papers in decent journals? Does that mean you have looked at grad level textbooks? – PVAL Apr 22 '15 at 23:44
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    And the question boils down to: HOW do I get in? What will help me more: Academics/Grades or proof of well roundedness/involvement/how good my application is. – Rellek Apr 22 '15 at 23:52
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    "there is a voice in the back of my head that keeps telling me that I only stand out because I am at a school that is not top tier, ivy league." As JeffE would say, Do not listen to the Impostor Syndrome! – chipbuster Apr 23 '15 at 1:53
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    The very very rare people who have been publishing since their teens (yes, they do exist) all imagine other people winning the Putnam in elementary school. Someday a precocious fourth-grader will win the Putnam; no doubt she'll worry about someone else winning the Fields Medal while still in the womb. – JeffE Apr 23 '15 at 2:44
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From what you have said about yourself, I think you're very well prepared for graduate school at a very good institution. There are always a few incredibly bright, very young mathematicians that fit the image you have in your mind of students at top tier universities. I know of a few myself. However a very large number of math graduate students are much like you (and I).

I had a very strong affinity for math as a young child but grew up in one of the worst academic environments in the US. A highly nontrivial amount of students in my area didn't graduate high school, even fewer went to college and fewer then even went to universities that were not the crappy local universities. I lucked out because one of the best high schools in the nation was in my area yet I didn't feel very challenged and more or less coasted along and didn't try. Ended up with a GPA of about 3.0 at the time of graduation. I even spectacularly failed statistics with a 40 or so from lack of effort.

Undergrad rolled around. I flourished in the environment: took lots of math classes, took plenty of courses outside my degree to broaden my horizon, did lots of research, but like you I was a big fish in a small pond. I felt similar to you - that I only looked good in comparison to all of the others. I almost had a 4.0 in math and had a 3.8 overall with a couple of years of research under my belt. When it came time for grad school applications, I cast a bit of a wide net, but got rejected from a lot of universities. (This was largely my fault since I seemed to mess something up on every application. Having not really done undergraduate applications, I was a bit overwhelmed and scatterbrained.) I actually got into my top choice university which surprised me a lot.

When I started my master's, I did notice that there was a bit of a gulf between me and the other students once I got there. My undergraduate institution didn't have nearly the course selection or resources that the other students had. However in terms of math capability, I was at least middle of the pack; so while I had a bit of a learning curve and some growing pains, I think I ended up being one of the stronger master's students because I fully dedicated myself and didn't give up. I did spend a ridiculous amount of time in my office the first semester, though. I went to school around 9:30 in the morning and went home at midnight many days.

For all of it, I grew incredibly and am doing quite well in my PhD program. Coursework is a cinch now, research is going well and I'm writing up a couple of papers which I hope to submit in the not-too-distant future. With your background and love for mathematics, I think you'll be in a much better position than I was in terms of applications. Your GPA is better, you seem much more fluent in many areas of mathematics and are much more mathematically mature. The biggest factor in determining your success (outside of pure genius - which is incredibly uncommon, even amongst mathematicians!) is your unwillingness to give up. Even if you're slightly weaker than some of your future fellow graduate students right now, once you get there, all of those disparities will quickly melt away if you put in the time and effort. Where you came from does not have to dictate where you end up. You are more than your past if you allow yourself to be.


Here is some general off-topic advice regarding grad school since you seem to be lacking in advising: just because you want to go to a top tier university and get in to one does not mean you should necessarily go. For undergrad, this is not the case; if you get into MIT, Stanford, Harvard, etc. and have the financial means, you should definitely go as it is a great opportunity and can directly impact your future. At the graduate level, things are so much more nuanced. (For example, the reputation of the university doesn't directly dictate your future success provided that you put in 100%. In the internet age, it is much easier to do really meaningful research and network with top researchers at any respected university since you have access to all of the information you want.)

When you apply, you are not applying to the university more than you are applying to a professor (or professors) or at least this is the philosophy I think one should have. You should have an idea of what kind of things you want to pursue for research and you should look to those people who do research in that direction. I don't mean to say that you need to know exactly the project you want to do but have a rough idea of the field you are interested in, say commutative algebra or functional analysis or harmonic analysis. If you apply to a university in which no one really pursues what interests you, you're going to be without a future advisor or you'll have to settle for second best. Applying to a graduate program just because of prestige doesn't guarantee success or happiness. Granted, at the top universities, this tends not to be an issue as they have very broad reach but it is something to be wary about. Just because it is a great university or great department does not mean it is a great fit for you.

There are also other factors to consider when applying to schools: Could you deal with the super competitive atmosphere (or alternatively the extremely laid back atmosphere)? Could you stand to live there for four to six years, e.g. if you're from a very hot climate, could you survive the very harsh winters in upstate NY or if you're from upstate NY, could you handle a Texas summer? If you cannot see yourself being happy with (or tolerating) a lot of these extra-academic aspects of where you are, you might want to reconsider.

A PhD is very demanding and can be soul-crushing at times. You'll often run into really difficult road blocks in coursework or research and if everything else in your life makes you miserable as well, you're going to have a really bad time. Your mental health is very important. You will be pushed to the extreme at times and in many ways throughout your PhD and your environment shouldn't amplify this. If you're having a tough time in research, the weather is absolutely miserable and you cannot stand your fellow graduates for whatever reason, your mental wellness might take a turn for the worst. I have seen this happen first hand and it is really unfortunate. These are not things academic advisors often tell students who wish to do graduate school because they are easy to overlook, but it is something to keep in mind. Hopefully this has been helpful.

  • This is a great comment. I honestly did not think about the extra-academic aspects such as weather. I really do hate the cold, and it seems like all of the top 10 schools are northern! And I like your perspective on applying to schools for some professor. I have not done enough research on this, since my books are all pretty old (Dover is cheap!) and I honestly have no idea who the big shots in mathematics are nowadays. I will definitely look into this, because I think you are very right about the importance of that. – Rellek Apr 23 '15 at 12:40
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    +1 for the last two paragraphs. those are certainly valid for other fields, too. – henning Apr 23 '15 at 18:33
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well I'm an MIT pure math PhD. I published my first paper when I was 27 : a couple of years after getting the PhD! Very few people has published anything before their PhD and their first paper generally was from their thesis.

If you have actually done some original research, write it up and get some advice from a professor at your school regarding publishing it. Being published is a massive help in grad school applications.

My general rule is only do a maths phd if you have a calling for it. It sounds like you do. I treated most of Grad School like a job -- go in around 9, take a break for lunch, go home around 6pm. Work hard during those hours. Enjoy your evenings and weekends.

High quality institutions do help your career so apply and if you get in, go and visit. If you don't like the atmosphere, go elsewhere. Certainly when I was at MIT the atmosphere was collegial rather than competitive.

  • _Being published is a massive help in grad school applications. _ - Would you say this applies to patents too? aren't they a kind of "publication"?. Obviously we would not be talking about mathematics (a patent in mathematics, that would be interesting), but engineering. – Keine Apr 23 '15 at 7:38
  • Thank you for commenting. It is nice to hear something from someone that is actually in a spot that I am yearning for, and this has made me feel much better about the situation. – Rellek Apr 23 '15 at 12:36
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Let me offer some perspective from an admissions point of view. What's important, for graduate studies, is your preparation, talent, motivation and work ethic. How will a committee assess this? If you go through a bunch of advanced courses with all A's from a top school, unless something is really wrong with your application, you'll get into a top school because the admissions people know the quality of the program you went through.

The main problem with coming from a small/unknown school (and I also did myself, so I'm sympathetic) is that it is harder to evaluate your preparation and talent, so to get into top programs there needs to be something in your application to gauge you against other applicants. Being able to do this is also helpful for personally knowing how you stack up with students coming out of top schools. Here are some suggestions:

  1. Even if you're not at a great school, you probably have some professors who got their PhDs from top schools. They will have a sense of how you compare to students from top schools. Consequently, they will be able to give you informed advice about what kind of schools you should apply to, and be able to write in their recommendation letters things like "Rellek is comparable to the better students I've known at at Ivy League Institute, Inc." Note: recommendation letters are particularly important when you come from an unknown school.
  2. If you've done things like summer programs with students from all over, your experience there can help gauge you against other students. (So it's natural to ask someone from one of these programs for a letter.)
  3. If you've written research papers or typed up notes on advanced topics, you can make them available (say on a personal webpage), so interested committee members have the option of taking a look.
  4. Consider applying to some backup schools and/or master's programs as backups. After a master's at a good school, you will be easy to compare with top students around the country.

As a final comment, it's true that to get into Harvard or Princeton, you should to be exceptional (or really lucky), but you merely need to be good and show promise to get into a top 10 school. I remember I was surprised when I found out most grad students I met at top 10 schools weren't "superstars" (some are, of course, but that's not par for the course).

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