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I'm currently an undergraduate mathematics student in my final semester. I plan to start a master's program in mathematics at the same university with an adviser that I did research with last summer.

I meet regularly with the adviser. He has stated that his goal is to help me go to a great PhD program, and the best way he sees to do that is to pump out publications, presentations, etc.

However, it seems that we are pursuing publication opportunities at the expense of understanding. He is giving me projects to work on that are frankly above my head. I spend hours and hours trying to find a bug in code that I didn't write about a subject I haven't studied. The time I spend debugging could potentially be spent more productively learning the fundamentals of our field. My approach has always been to understand a topic from the ground up, but he wants me to be able to work with PhD level code because it's the best way to get a publication.

I know that publications are essential. I want to eventually publish something, but not before I have built my own essential knowledge foundation of our field. Learning the topics thoroughly is more important to me than making a publication, and it seems to me that if one strives to truly understand and advance their field, then publications will come much more easily.

I plan on communicating this to him as clearly as possible. I have also drawn up a one-page document detailing my goals for my final semester as an undergraduate, my vision for my master's degree, and what I am looking for in the advisee-adviser relationship. I plan on delivering this document to him and using it as a guide for our discussion. Yes, I am willing to compromise on some points.

Of course, I understand that I can't just have everything my way. But I haven't started grad school yet, and now is the best time to cement the type of experience that I will have. As far as I can see, the cards are in my hands. If my adviser is adamantly against my approach, he can drop me, and I will go to grad school somewhere else.

Well, I wish I had a better question to sum all this up. What are your thoughts?

Edit: I am studying applied mathematics, computation, high performance computing in the United States.

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    "I plan on communicating this to him as clearly as possible." - Sounds like you answered your own question. – ff524 Jan 3 '17 at 23:11
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    Could you specify whether you are in pure and applied math, and in what country? What it takes to get into a strong PhD program depends on this... – Pete L. Clark Jan 3 '17 at 23:17
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    Well, hmm. If you want to get into a PhD in pure math in the US, then the advice "publish as much as possible" is really wrong, especially at the cost of neglecting the fundamentals. The farther you get away from pure math, the less wrong this advice gets. Most US math departments do not do separate admissions for pure and applied, and applied applicants are (I think) judged only moderately differently. But if you want to get a PhD in a department not called mathematics, then I'm a bit dubious about your supervisor's advice but am not really sure. – Pete L. Clark Jan 4 '17 at 3:22
  • Are there any other possible advisors in the department who might take a slightly less post-doc-ish approach to a master's? – aparente001 Jan 5 '17 at 2:26
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I'm a PhD student in pure mathematics, so I haven't been in your exact position, but perhaps something close to it. Nonetheless, allow me to play devil's advocate for a moment. (Feel free to take all of this with a grain of salt.)

My approach has always been to understand a topic from the ground up...

I think this is a great attitude. Being able to go from the foundations to the frontiers of research demonstrates a deep understanding of a subject. However, I think this approach is also wildly impractical as a young researcher. Ravi Vakil has written about a different strategy for learning, speaking of "tendrils of knowledge":

'Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards".' (Emphasis mine.)

(You can find a similar message in cartoon form here. Make sure to keep clicking on the images.)

Of course, there is a balance: if you really do not understand the basics of a subject, trying to dive into a research question will be fruitless.

I know that publications are essential. I want to eventually publish something, but not before I have built my own essential knowledge foundation of our field. Learning the topics thoroughly is more important to me than making a publication, and it seems to me that if one strives to truly understand and advance their field, then publications will come much more easily.

Again, wanting to learn for learning's sake is admirable. But remember this: You could spend your whole life (and probably more!) studying things that are already known without ever producing anything new.

All right, resigning my post as Lucifer's lawyer, I absolutely think you need to communicate your unhappiness with your advisor. Just make sure that you aren't overly confrontational. I think outlining your goals for your last semester and masters program is a great idea, but I'm not sure including a bullet-pointed "what I am looking for in the advisee-adviser relationship" is such a good idea. Other than that, I am glad to hear that you're open to compromise.

Some advice/questions:

1) Have you ever previously mentioned to your advisor your discontent with your lack of background? If you haven't said anything yet, for all they know, you're perfectly happy with everything, and this may blindside them a bit.

2) Ask your advisor if you can do a reading course on a topic related to your research. This way, you will have time to ask questions about the more theoretical aspects of the subject while still moving forward on your research.

3) Does your professor have a research group, or other students working on similar topics? If so, try turning to them for help when you have questions about fundamentals. Your peers are sometimes your best resource, and discussing things with your colleagues is sometimes the most effective way to learn something.

4) At my university, students have both a primary and secondary advisor. I think it's a great system: the secondary can provide another perspective, and can also mediate any potential advisor-advisee conflicts. Does your university have anything similar?

5) If you're sick and tired of debugging others' code, why not write your own? My own research is often very computational, and I honestly don't think I really understand an algorithm until I can implement it myself. That said, there is value in accepting some "black boxes."

  • A comment about the tendrils of knowledge and all that: please remember that this is Ravi Vakil's advice for current math PhD students. For that audience, it is excellent. [I know Ravi a bit, and I admire him as much as everyone else.] I think though that this is one of the biggest differences between how to learn pure math as an undergrad and how to learn pure math as a (maybe not absolutely new) PhD student. – Pete L. Clark Jan 4 '17 at 6:09
  • Speaking for myself, I got my undergraduate degree from one of the best and most "ground up" undergrad programs I know: Chicago. Really being solid on undergraduate-level algebra, analysis and topology was the best gift I could have given to myself as an early career PhD student: it helped me to pass my qualifying exams in my first year and move along to the tendrils. – Pete L. Clark Jan 4 '17 at 6:10
  • @PeteL.Clark Those are fair points. The OP is soon to be a grad student, so I think the "tendrils of knowledge" strategy does apply to some extent, especially in the more advanced topics relating to their research. I also may be a bit biased since I often have to fight back the thought, "Well, I should really read more about group cohomology/invertible sheaves/ample line bundles/central simple algebras before I try to work on this..." – André 3000 Jan 4 '17 at 6:26

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