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I am currently in my first year of a general mathematics PhD program at a state university that isn't rated particularly high. As far my classes go, I put in little to no effort towards them and instead dedicate my time to the research I've started, and still am at/near the top of my class. The students in my classes just seem to be at a much lower level of understanding than I am, and I am afraid I am stagnating myself by not being in a more competitive setting. I've begun working on a research project with a few other students who are soon to graduate, and thus far, everything we have done this semester has come from me, even though I haven't even finished the sequence of courses relevant to the material we are researching. Granted, there are a few professors who are well-respected in their field that I currently work under, and that is a great privilege, but I do not feel that most the other students there are my academic peers.

I would like to look into transferring to a better university, but I have one main concern; my undergraduate transcript is not very good. Upon close inspection, one would find that I received very good grades in the advanced classes I took, but poor grades in the easier classes I took earlier in my academic career. I got a 4.0 the first semester of my grad program, and am confident that I will 4.0 this semester as well, but will this be enough to get into a highly ranked PhD program?(Edit: This is a rhetorical question, not the question I am trying to ask. It is obvious that this is not enough, so I am trying to find out what outside of this I can do?)

So my question is should I look into transferring?(and maybe risk offending the professors I have a good relationship right now?) and if so how can I prove that what they see on paper(my undergraduate transcript) is not reflective of my abilities as a mathematician?

EDIT: I should mention that my biggest concern right now is that my classes are being held back by the people who don't understand the material. I would love to be in a class where things were faster-paced so we could get into more detail and more subjects. I've been studying into more detail by myself, but I feel I would benefit from seeing this from my instructor as well(which is what I would imagine happens at a university with a better student body), which I don't get because too many people in my classes are struggling with the basics.

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    It seems to me that your attitude might be something that could hold you back. I would resist the urge to think of your good qualities (e.g. your intelligence) as the primary factors contributing to your eligibility for a certain position. Your dedication and willingness to put in the time and effort to achieve the grades you need to get into the right school are just as core to your potential as your natural intelligence. Don't underestimate their importance! – quant Mar 2 '15 at 0:28
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    @quant being in an environment where you don't have to try at all doesn't give you the opportunity to learn about dedication and effort. – Kik Mar 2 '15 at 15:43
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    @Kik: If your grades in certain classes suck, as is stated is the case here, then not to "have to try at all" means that and getting poor grades. These classes likely test a different skill set (eg, calculations emphasized over derivations), and to dismiss poor grades essentially as a result of classes being below oneself is orthogonal to dedication and effort - it reeks of hubris. Also, "more advanced classes", if seminar type, could whitewash ability and allow you to excel by a strong personality only. – gnometorule Mar 2 '15 at 20:12
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    @gnometorule Alternatively, the classes could put undue focus on homework over tests, in order to force people to do homework, hoping they will learn "better" that way. Unfortunately, those who do not need to do homework assignments to understand the material and ace the tests have their time wasted, and may decide not to do the homework at all, leading to poor grades, despite a better understanding of the material. This was indeed my own educational experience from middle school through high school and college even. – Kik Mar 2 '15 at 20:27
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    @Kik you're absolutely right. It's not the OP's fault they are being underwhelmed at university, but it is going affect their ability to progress, so it's best to be aware of the consequences and try to work with them. Dwelling on the causes of the situation is unlikely to help him/her progress. – quant Mar 2 '15 at 22:01
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Many, many students who are at lower-ranked universities aspire to be at higher-ranked places instead. (In fact, what is more surprising to me is that not all such students feel this way.) The hard truth is that top-ranked universities by definition are very selective of their applicants, and in your case the "main round" of the selection process has already occurred. It is not impossible to transfer to a much better place, but it is difficult, to the extent that it may or may not be a reasonable goal. Growing where you are planted sounds like a smarter strategy to me, and in fact it is not fundamentally incompatible with the ambition to move elsewhere.

The one way I can think of to effectively demonstrate your potential is to realize it, i.e., to prove at least one theorem of a caliber beyond what most students in your program do as part of their PhD thesis work. If you can do that, then you clearly deserve to be at a better place, and you should not have much trouble finding some place to take you on. I don't think anything else is worth banking on: as others have said, grades in PhD-level mathematics courses in the US are not necessarily very meaningful. This varies in its extent, but every program I know has enough courses where all students who do not do anything egregiously, ridiculously wrong get the highest possible grade that a 4.0 GPA at the graduate level is just not the same thing as it is at the undergraduate level. I would expect that a substantial proportion of the students in your program would have 4.0 GPAs.

Unfortunately I think the strategy of "gunning for an early research success" will either be a good one or (and in terms of the percentages, this is more likely) a very bad one. Most early career math PhD students are not anywhere near equipped to do meaningful research, to an extent that other academic fields would have a hard time believing. In my program (about the 50th best in the US) early career students neglecting their coursework in favor of their "research" is a big red flag for me: these students often fail to meet programmatic requirements and get in real trouble.

To be very honest: I think the big risk here is that you are not as exceptional as you think, that your lack of effort in your coursework will itself become a problem down the road, and that your superiority to those around you may only be a matter of opinion. Of course I'm not saying that this is the case -- I couldn't know -- but it would be very smart to be concerned about this on your own behalf. You need some reality checks. Here are some checkpoints:

Step 1: Can you pass all your generals / quals / required coursework rapidly and unusually well?

In most non-top math PhD programs I know of, there is an initial phase of 1-3 years where students spend time studying coursework and then passing exams. A small number of students get through this portion much more quickly: within a single year, a single semester or even immediately upon arrival. It is debatable whether such students are truly more talented than the others or simply arrived very well prepared and have their act together, but in any event such students are certainly doing exceptionally well in the program.

Step 2: Can you get faculty on board with the plan of your having exceptional promise and ambition and get a very meaty project to work on?

Go to one of the faculty members you're working with and level with them: say that you feel understimulated by the program, are worried that your peers lack your intensity and ambition, and that you are really looking for a project to sink your teeth into. Then step back and listen very carefully to what she has to say. You will probably get some amount of corroboration/denial of the picture you have painted, though it may be subtle. (I would not say: "No, I completely disagree: you're just average, maybe not even that" to a student, even if I felt that way!) More importantly you should get something to work on. Do so. If it is not what you wanted, or if you have your own ideas about what you want to work on, come back and say so, but I think it's good to spend some time on projects that other people give you at this stage.

Step 3: Can you get some results? Anything at first, then something nice, then maybe a real breakthrough?

If you can make it successfully through Step 3, then as above you're ready to go somewhere better. Let me not even try to hide the following key point: the merit of this checklist is that it's what you're supposed to be doing anyway. If by the time you successfully complete Step 3 you're ready to graduate, should you curse my trickery? No, you should proudly write up your thesis and expect to get a postdoc at a much better place than you did your PhD, which is hard to do and puts you on a great trajectory for your later career.

Good luck.

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    From what you are saying, it sounds like I should be somewhere else. I am on a 3-year track right now. I will be taking two of three prelim exams this summer, and the third after completing the relevant course sequence next year, so it seems like I'm doing fine on step 1. As for step 2, I am the only first year student in my field doing a research project. My professors have noticed my potential and have commented on it. For step 3, as I mentioned, I am the only one producing in my research group, and I have made short work of many of the topics we are exploring. – SE318 Mar 2 '15 at 0:46
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    SE318: The point is to do the steps in order. I'm glad you're doing well, and again I don't know you. If I wanted to play devil's advocate, I'd ask: if you are spending so little time on your coursework because you don't need to, why aren't you able to take and pass all of your prelim exams at once? Also, what are you producing in your research group? Publishable results? If so, great. – Pete L. Clark Mar 2 '15 at 1:18
  • I would love to take all my prelims at once and get them out of the way. I didn't think that was an option though, as I thought I had to take the required sequences of courses before taking the prelim. I will look into that to see if it is actually necessary to take the courses prior to the tests. – SE318 Mar 2 '15 at 1:47
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    @SE318: You have a great many things to talk to your advisor about. In particular, the idea that you still need to do the coursework even if you show that you have mastered it: I doubt it, but it's not up to me. But yes, passing your prelims rapidly and well would give you some consideration in transferring. "I think my research is publishable(how can you tell if it is?)." Talk to your advisor. – Pete L. Clark Mar 2 '15 at 2:22
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    what is more surprising to me is that not all such students feel this way — there can be many reasons to not aspire to move. People can have many different personal reasons to prefer to be at institute X even when institute Y has more esteem. – gerrit Mar 2 '15 at 5:18
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will this be enough to get into a highly ranked PhD program?

No.

You might well be able to get into a highly ranked PhD program, and have your bad undergraduate grades overlooked.

But this is difficult; many more try to do this than succeed. Perfect grades in grad school are common and will not make a strong impression on graduate admissions committees. What will be more impressive is going far beyond what is expected of you, learning advanced topics on your own initiative and discussing them with faculty, being a leader at your current university (e.g., helping others, giving seminar talks), and carrying out research. If, next year, your rec letters read "SE318 is only in his/her second year in the Ph.D. program, but his/her research output already exceeds that of most of our graduating Ph.D. students, and he/she clearly deserves a shot at a much stronger program"... then you have a decent shot.

In particular:

I put in little to no effort

If you hope to realize your potential then you must start putting in much more effort immediately. Outwork everybody and be someone that everyone looks up to.

Best luck to you.

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    I appreciate the feedback. I should clarify, when I said I put in little to no effort, I am talking only about my effort specifically dedicated to my current classes, not effort dedicated to my studies as a whole(E.G. I am dedicating most of my scholarly time to the research I've started rather than to studying for my classes as even without studying that specific material I am still outperforming most(maybe all?) of the other students in my classes. – SE318 Mar 1 '15 at 21:36
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    @SE318 if you want it to show, you probably have to shine consistently across everything you do. Grades are a very easy thing to look at. – Davidmh Mar 1 '15 at 23:52
  • Put in effort in your current classes as well. Do you have qualifiying exams at the beginning of your second year? You can often pass these with 60% or so, but try to ace them. – Anonymous Mar 2 '15 at 13:08
  • @anonymous I do have two exams scheduled for this summer. I am confident in my ability to ace both of them(and plan to make sure my confidence is well founded). – SE318 Mar 2 '15 at 16:20
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You might find your attitude holds you back. Universities are much like any large organization, and not like movies such as A Beautiful Mind or Good Will Hunting, initially you are little more than a name on a list. If you want to be noticed and get involved with exciting research, you have to put in the time doing the tasks the professors have set you; if you don't you may find you're considered a slacker, or worse, insulting the teaching staff by implying their work is beneath you. To get anywhere in academia (or indeed business), you need people on your side, and if that means sucking down your pride and getting the job done, then so be it. If you're ambitious, it will be noticed because your excellent grades will stand out to those who have the influence.

Richard Feynman was given permission to study other subjects on top of physics, as long as he did the work for all the subjects - this he did with grace and time to spare. Shine by your diligence and hard work, and it won't matter much which institution you're at, your work (and association with those eminent professors) will do you the most good. Besides, you may find you actually learn something along the way.

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The answer is very simple. You have realized that the courses are easy and getting a high mark is not difficult. So you should not expect this will be selling points for you to get into a top program. If you know this, then the admission committee from Harvard University must know it as well or better than you are. Otherwise, what is the point to go to Harvard?

At this point in your career, it is much more important for you to have a solid foundation in whatever you are interested, and do not plunge into research right away. The first two years of a typical PhD program is to help you consolidate your understanding of the material. If you skip it and regard the courses at face value because you can easily get high grades through easy psets, it would be more detrimental to you than good in the long term. This is to exclude the chance that you are very talented, independent, hard working students like John Nash, Samuel Donaldson, Alexrander Gronthedieck, etc. But even they would need an advisor.

I will suggest you try to read some faculty's papers in your department and find some potential advisors after you pass the generals. Transfer only make sense if you can work with a much better advisor in some other university, or you want to switch your research field to something simply not available in your department. But even in this case it is very risky bet.

As a fellow math PhD student, I am not as experienced as Prof. Clark or others in the forum as I have not finished my PhD, but I do have 3.95/4.0 GPA in my grad school, fastest graduate student on department record to pass the qual, etc and I have went through a similar confusing period about my identity and my future. So this is my sincere advice.

Best wishes for your future academic career.

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What you learn in graduate school doesn't just come from the professor. You are there to learn how to learn for yourself. After you finish your Ph.D. and move on to some academic position there is will not be a professor continuing to teach you new things. You need to learn how to teach yourself if you want to advance. If you are being graded on a curve then a high GPA does not mean you understand the material. It means you understand the material better than your peers. Even if there is no curve, then the professor may be reducing the difficulty of the exams to fit the ability of the class. Either way, saying you don't put in any effort because you outperform people who don't understand the basics does not suggest that you understand the course material.

When you find the other students are struggling, see if they want to form a study group to go over the material outside of class. Do not tell them you want to form this group because you think you are brilliant and everyone is holding you back. By studying the material with the rest of the class you will be forced to put in the effort. You will be forced to examine what you claim to know and you will see various misconceptions you have about the material. Misconceptions which won't keep you from being at the top of the class but will keep you from having a true understanding of the material. You will learn things from the other students that are at/near the top of the class. You will gain teaching experience. You will make some friends. You will help to bring the quality of your entire program to a higher level.

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    This is sound advice, but it does nothing to answer the question. I want to transfer. I believe I can succeed in a better university than the one I am in right now, so I want to know what is the best way to facilitate that considering that I have a bad undergraduate transcript. Sure I could help bring the quality of my entire program to a slightly higher level, but that isn't the point of working on a PhD. I want to get the best education I can right now, not give others a better education. If these exams are reduced in difficulty, then I need to be in a place with a more difficult program. – SE318 Mar 3 '15 at 17:28

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