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The problem is pretty much the title. I have a bachelor and master degree in applied math. I have been assigned a small problem to start with, from a professor (who is considered very good in my field) who is willing to give me a reference, which in turn I need in order to apply for a PhD position. Our common goal is for me to work gradually on the problem and, if I solve it, produce a published paper. This is a huge matter for me as do not yet have any publications, and it will (I believe) boost my chances to obtain a PhD position. The problem is that I have been working for a month on this and I have not reached an answer. So, considering that he said that this first problem would be pretty easy to deal with, I am really wondering what to do.

I need an answer on this: Should I admit to this person that is highly accomplished that I am having difficulties at what he asked from me? or should I tell him all I have got up till now even if that is not even close to a solution?

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    Uh, both? Tell him what you did so far AND tell him what you're having trouble with. – ff524 Jul 13 '16 at 15:41
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    It's not a question of ego. In most collaborations, no one expects you to meet next time with the finished paper: you meet to discuss your progress, and take it from there. If this can really lead to a publication, it's probably less obvious than you seem to believe it is. – gnometorule Jul 13 '16 at 16:06
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    Of course tell your professor. If it turns out that you could not do that problem, he may adjust it or propose a different one to start on. Even if (unlikely worst case) he says "If you can't do that then you shouldn't be working with me." it is better to find out now rather than later. – GEdgar Jul 13 '16 at 16:09
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    I must add that a month it is a very little time to work on a problem. Just be fair, tell you professor what have you done so far. Also, you can try to discuss different problems or approaches with your professor. – Mikey Mike Jul 13 '16 at 16:14
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    Based on your other recent question about your career in math (posted today) and that you just joined (today), I think you might be a little worked up (today). You stated in your other question that you've only been looking for PhDs for 3 months, and you've only been working on this problem for 1 month. I've spent a month working on just a couple lines of code before. Leave your ego behind, take a day off, and talk this through with your advisor. – Hobbes Jul 13 '16 at 16:19
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I don't know why you think there are any options other than both. Of course you shouldn't just say "I'm having difficulties, full stop" without talking about what you have tried. Of course you shouldn't just say "this is what I did" without also talking about where you're stuck.

You sound like you're worried about making the best impression on this professor. The best impression you can make here is showing that you have the maturity to deal with doing research: to work hard on a problem, be aware of both what you've accomplished and what you haven't, and be able to ask for help where you need it.

There are bunch of things that could be happening:

  • You went off in the wrong direction, or missed something, and the professor will be able to help get you back on the right track.

  • The problem is actually harder than the professor thought, and explaining why you're stuck will make that clear to them.

  • The problem is harder than you thought, and the professor expected you to need some help.

In all of those cases, the absolute best thing to do is honestly present the work you've done. The less you worry about framing things to make you look good, the better you actually end up looking.

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    Good answer. As OP says in another question that they are looking at industry positions too, it's maybe even more important to lose this attitude for industry. There's nothing more toxic than a lone wolf. Always rely on all resources available, and share them (progress) freely with others. – gnometorule Jul 13 '16 at 16:15
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    Two more bullets: the professor couldn't solve it but thinks (or hopes) that you can; or he knows it's not solvable and is testing whether you'll admit it. – WGroleau Jul 14 '16 at 7:53
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    Also, the process of research is collaborative. It's very rare in math research (and probably everywhere) for someone to just receive a problem and solve it. You have ideas, bounce them off your advisor or collaborators or friends, receive their thoughts and ideas, read papers, repeat.... You don't just flex your genius muscle and extrude a paper! But don't worry, you wouldn't know this unless you get started. – Richard Rast Jul 14 '16 at 12:06
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    Another possibility is that the problem the professor thought he gave you isn't the same problem you've been working for a month on. Perhaps the professor was thinking of an easy case, while you're trying to solve the (much harder) general one... – poncho Jul 14 '16 at 13:32
  • +1 for the last sentence " The less you worry about framing things to make you look good, the better you actually end up looking". – user21264 Jul 17 '16 at 7:47
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I am a professor of (theoretical) mathematics at a state university in the US. In recent years I have advised a number of graduate students: so far, one master's student and three PhD students have written theses under my direction, and I currently have two PhD students.

Just now I tried to look back over these six students and recall if any of them ever just solved a problem I gave them with no intermediate discussion or help from me. This definitely happened once: one of my students did something brilliant over the course of a couple of weeks that became the main core of his PhD thesis. There is another case where a group of students working in a research seminar with me quickly and completely independently came up with a clever, computationally-intensive way of answering a question soon after I raised it in the seminar. And my first student was exceptionally strong: to my mind his thesis work was done mostly independent of me, but that did not stop him from asking a steady stream of technical questions both of me and his co-advisor.

I could complement each of the stories above by half a dozen more involving the same students, in which they got very substantial help from me, or -- even more commonly -- simply abandoned the problem because they did not have the prerequisites / could not make progress / was not to their taste / was too difficult, too technical or otherwise poorly chosen by me. I won't tell such explicit stories, of course, but you should be aware that virtually every graduate student in mathematics could. In my opinion, thinking that you need to talk to your advisor only after you've solved a problem (even a "small" one, or even a single step of a problem) is a major misunderstanding of what the student / advisor process is like in mathematics. I don't know how it is in other academic fields, but as a student in mathematics when you start (trying to) do research you don't have a clue. Somehow you need to get from not having a clue to research success in the span of several years: you do that by getting a lot of help from your advisor.

When I was a PhD student (at one of the top programs in the world, with one of the most eminent advisors in the world) I was very independent. I would usually meet with my advisor less than once a month. When I met with him more often, I felt like I was telling him an incomplete story: you asked me about this, and I am trying this, and it is starting to look like it won't work, so maybe next I will try that...I wanted to go through the entire process of acquiring background and trying to solve a problem in all the ways I could think of by myself: by talking to him sooner than that, I felt I wasn't giving him my best effort. In retrospect I really think I played this mostly wrong: my approach fanned the flames of my independence, but at the cost of much of the help and insight I could have gotten from my world-renowned advisor. In recent years my professional confidence has grown a lot, and I am much more willing to tell someone "You know, I thought about what you told me [for, say, a day or two] and couldn't work it out. Here's where I got stuck. Can you tell me a little more about...?"

Now I work with PhD students and often wish they would check in more often -- both more often chronologically and more often in terms of steps of their own thought process. I've had so many meetings where it turned out that students were spinning their wheels for 4-6 weeks on something that gets cleared up immediately upon meeting: either I resolve the point they're stuck on (there is no shame in that, by the way; I would much rather a student spend three months on a problem, getting stuck every so often and allowing me to unstick them than spending three years solving the problem completely independently) or they misunderstood what I told them and are going down the wrong track or it turns out that what I suggested definitively doesn't work.

The point is that getting help from your advisor -- substantially and often -- is much of the value gained from being in a PhD program at all. You should not at all feel embarrassed about asking your advisor for help. Instead you should work on asking for help in a way which shows knowledge and professionalism: don't just say "I'm stuck." Explain where you're stuck. Better yet: when you leave any meeting with your advisor you should have been given at least one specific thing to try out, and when you come back you should report on that (very small, usually) thing. If it worked...but seriously, it usually won't work, in which case you should endeavor to try to explain why it didn't work. Just coming and saying "I couldn't do this" to your advisor is not very helpful. At best, it is liable to elicit your advisor telling you what she thinks will work to solve the problem: this could be helpful...or she could solve the problem before your eyes.

Let me finally say that most thesis advisors I know have relatively poor ideas about the true difficulty of a problem and how long it will take a student to solve it. (To be fair, you can give the same problem to two students of ostensibly equal background and abilities, and very often one of them will spend months or years longer on it than the other.) An advisor who thinks that a problem is "quick" or "easy" probably means that it is quick or easy for her. Also, there may well be (and arguably should be, at least eventually) unforeseen difficulties. Finally, academics are famous for underestimating the amount of time it takes to do anything: the same advisor who is looking down her nose a little bit at you for taking several months to solve an "easy" problem may well have just sent an email explaining that her referee report will be several months late.

Good luck!

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    you don't have a clue -- If you had a clue, it wouldn't be research! – JeffE Jul 13 '16 at 18:33
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    What a wonderful answer, very thorough. As a side joke, I would feel like asking what's the difference between "mathematics" and "theoretical mathematics" :-). – gented Jul 13 '16 at 20:44
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    And "easy" may mean different things. I think of things I could do in months, rather than years, as being "easy problems." – Kimball Jul 13 '16 at 23:25
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Even though quite good answers to the question were provided I will share my story as I hope some people might find it insightful. I am a PhD student (applied math, mostly quantum related stuff as I did study theoretical physics before) just finishing the first year and what I will share is what happened to me for the past few months.

During a conversation with my advisor (and I guess it was not even proper meeting, but just some talking in the corridor of the institute) she told me that I should probably try to prove this one thing, that it should hold. I did that quite fast as it really was very easy. Then I did study some literature, mostly recent papers and I realized that probably even the converse should hold. As it usually goes, this was way harder to prove, but I was unable to find a counter-example, I was capable of developing some idea how the proof should go almost immediately but for a week or so I was not capable or putting together the actual proof.

When I did talk to my advisor during our weekly meeting I did present my idea on how the proof should go based on some assumptions and I admitted that until now I was not able to prove the result properly. Since this was a problem I did come up with myself she did not have an idea how to prove it, but she pointed out that I should look up exposed faces in the literature. Literally, she told me something like: "Look up exposed faces in Barvinok's book". Just that. Even this simple sentence helped me to finish the proof (if it is correct, I have to wait until august to ask her about that).

The morale is that you should talk to you advisor no matter what. In my humble opinion the reason is that those people after working in mathematics for years have a lot of experiences and can point out useful things even if they can not provide the full answer.

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Ok, not gonna lie here...I have degrees in RTVF & English, and math set off my panic response in my lovely little anxiety bubble I hid quite well (with the exception of people trying to tutor me in math - there was no hiding it then.)

I do, however, see parallels to my writing efforts to get published, gain traction with those for whom my career was being nurtured, etc. I apologize right off the bat if this is terribly simplistic or irrelevant to your quandary! "Occam's Razor Theory" popped into my head...& all I can think about are all the times I went through the looong litany of ways to achieve my desired end result, always with that notion that at I could quit, when all along, I was missing the point - simplicity is the answer to overthinking.

Everything I'd ever learned from my parents AND the best educators I'd been fortunate enough to have had take me under a wing. That advice was ALWAYS when struggling, seek out that person for whom the work is to be validated, or that mentor somewhere that can help you refocus, etc. Because it's not a bother to them...

No matter the level you ultimately land in a career, there will always be need for collaboration. No one truly goes it alone. And from what I have heard about such things as your current work, there are those who take months if not years to complete - if this is not one of those kinds of taskers that allow for that allotment of time, all the more reason to go to the person in the know for help. It will not be a blight on your abilities, rather a star on your wisdom in doing what needs to be done & getting that info from the right person. (Let's face it, the "right person" getting asked for a bit of help never insulted that "right person!")

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I would like to to add to the previous answers by offer one extra piece of advice. Before admitting to your advisor that you are stuck, try to find some tangentially related problems and make a little bit of progress on those. Your advisor will be happy that you have something, and it also shows initiative. It will possibly give your advisor more ways to direct you in the future.

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    It's a bit like the general requirements for asking on Stack Exchange sites. Show effort and research of your own and help will be given freely. – Trilarion Oct 26 '16 at 8:34
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The answer by Pete Clark is beautiful and kind and reflects a lot of real in the trenches time. I will just add a little perspective (more from the worker side, not the manager...never been a prof but did my "union card").

First: Realize that the incentives of professors and students are different. A tenured prof in a research setting wants to solve dramatic problems that get him prizes. These are butt-hard. There are profs I know who have burned multiple students in a row having them bang their heads against a problem that they didn't solve (in some cases wasting years of their time and not getting the union card). For the student this is a calamity. For the prof, it is not. Now, I absolutely don't think your prof is doing this. But you need to realize it is an extreme example to show the difference of student and prof.

Second: I have learned doing a lot of new, hard problems as a consultant that it is easier to be stupid early. A month is not the end of the world (for an academic) but it would be a calamity in the business world. In general at the beginning, you want to interact and show something, progress, effort, difficulty, etc. There is a slick way to show it ("here is my progress" versus "here I am stuck"). But in any case, it is better to interact early and get redirected early than later. This is not just an academia thing, it is a life thing!

I actually do have a huge value for the Message to Garcia (read it). You don't want to have no initiative. But at same time if you are stuck...interact early.

0
Our common goal is for me to work gradually on [a small] problem and, if I solve it, produce a published paper.

A published paper pretty much implies a problem not previously solved. "small" in that context means something like well-confined. However, the well-confinedness is a guess. Fermat's Last Theorem looks like something that should be doable with congruences, and pulling something off with congruences is a lot of routine and skill. Small problem. Now it has been proven that within a given framework of congruences, Fermat's Last Theorem is not provable (not false, merely not provable). This proof ruling out a whole infinite class of "small problem solutions" is, in turn, not actually a small problem.

It turns out that even though FLT seems very much in the ballpark of a lot of "small problems", it's actually on a very slim borderline of "true by a very small margin", requiring a whole lot of new techniques to reel in.

You haven't reeled in your fish yet. Nobody else apparently did either which may just mean that it's a fish in a boring lake nobody visits. But it can still be a big fish whether or not it would be edible.

Maybe you just need somebody to show you a different angle of pulling the line. Maybe you need an entirely different reel. Maybe the best thing you can hope to do is to get a good size estimate of the fish to alert bigger fishers.

Your have an advisor, not a task master. This is a perfect point to get his advice. As a researcher, you are expected to spend your time well. When you stop making progress, you are not doing so. Now sometimes you just need to get stuck in a certain place for long enough in order to find a solution. But that's expensive, and you should make it count. Getting stuck somewhere where a more experienced person can point you out in 5 minutes is not making it count. You may or may not be there.

So: discuss.

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