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I'm a first year student undergraduate at a top US university, and I'm going to major in math. Of course, I still don't really know what I want to do with my life, but I'm really excited about going on to a PhD in math. What does one have to do to get into a top PhD program in mathematics? When I say top, I mean absolute top - like Princeton or MIT. Please understand that I'm not being presumptuous in asking this: I don't know whether I'll ever be good enough to do a PhD at an institution such as these, but I'm just asking for reference - and out of curiosity.

I've read a lot of threads about similar topics, but answers there are fairly vague ("good letters of recommendation", "advanced coursework", etc). What exactly do these terms mean, and what should I - as a first year student - already start doing to at least stand a chance sometime in the future to even dream of being in a program such as the ones I mentioned?

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    Get good grades and apply when you have finished final year... – Solar Mike Mar 28 at 6:28
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    Let me be blunter than I or most mathematicians would like to be. There is such a thing as mathematical talent, and, starting where you are and given the level you're aiming for, it matters (though many other things also matter!) Let me make an analogy to making the NBA. One could make a list of things that most basketball players have to do to make the NBA, but almost all of the basketball players who try to do most of these things still won't make the NBA. In fact, so few of them will make it that calling such a list a guide to making the NBA would be ridiculous. – Alexander Woo Mar 28 at 8:00
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    I think a lot of it has to do with opportunities taken at a young age. Enrolling in a program to skip part of high school and start college early is an example. Basically, one initial requirement would probably be access to math beyond what one typically sees. Another example is participating in the Ross program. Or working with highly motivated students on challenging math. If you can forgo everything and focus on math, that would probably help. Go to a top college and take as much math as possible. It helps to have mathematician parents. – user74089 Mar 28 at 9:05
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    @AlexanderWoo The problem with the NBA analogy is that there are much clearer ways to measure talent and performance in basketball than in mathematics. Even if "mathematical talent" were a one-parameter quantity, it's more than a little dubious that Princeton and MIT are particularly good at identifying it. – Elizabeth Henning Mar 28 at 17:11
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First off, I have a PhD in a top 20 - not top 3. I believe my school was 16 or 17 when I was doing my PhD. So I'm below what you are aiming for. So feel free not to read this post.

First of all, I'm puzzled by your comment:

fairly vague ("good letters of recommendation", "advanced coursework", etc). What exactly do these terms mean, and what should I - as a first year student - already start doing to at least stand a chance sometime

Good letters of recommendation: You will need to get about 3 profs to write good letters of recommendation when you apply to grad school. That means you need them to say you are smart as in PhD-smart.

Advanced coursework: Take ALL the higher level courses if possible. Take some graduate level courses if possible.

Now about grad school: don't think about the top 1 or 2 or even 3 for now. If your goal is math, then your goal is math, not the school. Just focus on math for now.

I do not know you, so whatever I said below might be useless and meaningless.

What can you do as a first year student? Get a good GPA, as close to 4.0 as you can. You want to take as many math courses as you can right? So do some self-study and test out of your gen ed courses. I had a student who tested out about 8 so that he can have time to take ALL the higher level courses.

Are you taking any math courses now? Then focus on them and do well. Doing well does not mean just getting an A. It means being near the top of the class. Your prof knows. Not all profs will take note. But some do.

You have to figure out which profs are interested in growing PhD students. Profs who care will be open to chat with you and guide you. The only way you can catch their attention and learn from them is to be at the top of your class. Ask questions in class (if possible, when appropriate). Do you study ahead? Talk to them during office hours and ask questions so that they know you are interested and are studying your textbook ahead of the class.

When I was an undergraduate I always study ahead. While studying all the courses, I will pick one book and study ahead until I'm done with the book for that course. Then I'll pick another one and study ahead. Etc. Don't just read the book. Do the problems.

It's also a good idea to think about your favorite area in math and study it on your own. Unfortunately if you are not very deep into math, you might not know where to go with this. That's why knowing a prof well will help. He can guide you.

If you have time and if you have not done so, I suggest you look at the textbooks for math olympiads. Study them and do all the problems. Try to finish as much as you can so that you have some time to try some putnam competition books. Math problems in the math olympiad and the putnam are very different from the type of problems you will solve in your regular math classes. In many ways they are closer to research-type math problems. Again, I do not know you. Maybe you have already done lots of math olympiad training.

Another very important thing to note is that it is your responsibility to keep your level of interest in your area (i.e. math) as high as possible. That means spending some time reading up on the biographies of famous mathematicians. Don't do too much of that since you do have to study math.

Books are the most important resource for you right now. Ask your profs for good recommendations. Do not be surprised that your class textbook might not be the best textbook on that subject. It's just one that's convenient and easy to use. I have no idea where you are in your math education. But if you are in Calculus, then you can for instance study "Calculus" by Michael Spivak. If you are very strong in math and have already studied that, you can go on to other books.

Check your math library and see if you can find magazines you can understand. Try the American Mathematical Monthly and the Mathematical Intelligencer.

See if there's a math club you can join. Make sure it's a math club and not just a social gathering for math majors. Nothing wrong with socializing, but if the club does not have math related activities, then it won't help your goal.

If you work very hard in the first 2 years, you might know what area you want to go into. (But it might change.) And if you take some grad level course(s) in that area in your third year, you might be able to attend the research seminar in that area, which might be a weekly meeting. You might be able to start doing some research during your senior year.

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    This is good advice except the part about math competitions. Those questions are nothing like research problems. – Tobias Kildetoft Mar 29 at 5:43
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Roughly: good grades (3.8+ GPA) in difficult courses, good test scores (80+ percentile on math GRE subject test [not the regular GRE math, which you should get a ~perfect score on without studying]), strong research background and good letters corresponding to it.

That will get you into schools in the top ~30. To get into the very top programs, you will need to meet this standard and also have either (1) a very nice research background / letters, or (2) something "interesting", such as impressive accomplishments outside of math

Disclaimer: not a mathematician

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    Can the downvoters explain? Not a mathematician, will delete if I'm off-base, but I got into a top-10 physics program with basically this formula... – cag51 Mar 28 at 5:52
  • I understand that. What exactly do you mean by accomplishments "outside of math" though? What kind of accomplishments? – gtoques Mar 28 at 6:18
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    Voted this up - can't see why it gets downvotes... – Solar Mike Mar 28 at 6:50
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    This advice is far too generic. Good grades (3.8 GPA)? There is no mention of the caliber of courses. Median Princeton admit probably has a 4.0 GPA with stellar performance in numerous graduate level courses like differential geometry or algebraic topology. 95+ percentile on GRE is probably a given although I doubt top programs put much weight on it at all. Most importantly, it really helps if a respected mathematician gives an absolutely glowing recommendation regarding your research potential. Impressive accomplishments outside math count for very little. – zoidberg Mar 28 at 7:06
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    @zoidberg surely the advice has to be generic - the OP is nowhere near completion, has no grades to speak of and not applied to any institution yet.... – Solar Mike Mar 28 at 7:14

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