working on an open problem for the master's thesis, without any guidance or help

Question

I am a master’s student studying mathematics (from Germany), currently writing my master’s thesis. In total, I have around six months to prepare, write and finish my thesis.

My advisor gave me an open problem to work on and he does not expect that I will solve it, but he expects that I am able to make new contributions on it, so that I have something new for my thesis. However, many established researchers in this research area are trying to attack this problem as well and for many years, but no one was able to proof even special cases or intermediate results until now. And my advisor doesn’t really know how to attack this problem as well, and this is one of the reasons why he does not guide me.

Already at the very beginning I told him that I’m pessimistic to be able to do this, but I just wanted to give it a try and I didn’t know that I have to work completely on my own. Now, I have tried it for two to three months, and all my attempts failed and my ideas didn’t work. Furthermore, my professional knowledge is still very limited, meanwhile I hit the wall and don’t know how to proceed.

I literally explained my advisor that I need his help and guidance to make some progress and I have told him all my attempts. The only reaction is always: He doesn’t know it either, and every time he just gives me another open problem (sometimes liked to the old one), without any idea how to attack these other problems. After trying the other suggestions, unsuccessfully, I decided to work on the first problem again (but still without any progress). I asked him again for help, and I literally explained him my situation that I'm stuck, and giving me another open problems is not that type of help which I need, what I need is guidance. But still no help.. Since I have to hand in a thesis this summer and there is no possibility to change the advisor, I really don't know what to do...

Do you have any idea or suggestion what I can do, or do you have similar experience? Or if it’s too opinion-based: what are possible ways to get out of such or similar situations?

Answer 1

First of all, your English is quite good, but as a fellow German I am well aware of the jokes about Germans excusing their poor English skills in high level prose. xD

Now back to topic: I myself had to deal with an open problem during my master thesis, so here are some points I learned during that time that might help you: (Disclaimer: As I wrote my thesis somewhere in between linear algebra and algorithmic number theory, some points might slightly differ depending on your field).

• Mathematicians are only human. Yes, after sitting in lectures, with professors telling you all the brilliant results, it might seem as if a professor, someone who finished his studies in mathematics, might be some omnipotent entity that is always right. But that is not the case. That the problem you are working on hasn't been solved yet does not mean that it is impossible for you, it just means that the people trying it up to this point where not capable to do it. Maybe they hadn't enough time, maybe they hadn't gotten the right ideas, maybe it just wasn't tried by so many yet. It seems to me that you are expecting your advisor to know the answers and just withhold them. But I'm rather sure he does not, he just doesn't know and wants to learn about it.
• Even if you can't solve the problem, you can develop methods to help with it. For example you might implement efficient algorithms to compute many, many examples, which might then help you (if you have the time) or others to see a pattern and solve it. I don't know your topic, so I don't know if it would be best to use algorithms, plots,... to show results, but I'm sure you can come up with something.
• You stated that your advisor does not expect you to completely solve the problem. That is in fact good. That means that even smaller contributions might be helpful. Can you get result for certain subclasses of the problem, can you solve some cases that are easier? Can you give relations and reformulations, e.g. "Problem A is equivalent to problem B, assuming condition C (thus I devote a whole chapter to the question when condition C holds)".
• Find structural results related to your problem. For example say your problem is of the form "We have structures of type A and want to find the automorphisms between them" and you are not able to find these. Then it might still help to study the structures of type A, find other ways to see/write them, get any results on them that might then help someone else to find the automorphisms.
• Prepare a time plan and discuss it with your advisor. For example you could say "I try to find results on problem A for another two weeks, if I don't find anything I will write down results on the structures (to at least have something to present) and after that look at these special cases: ... At (date) I will start writing down what I have up to this point, no matter how much it is." If you have a well worked out plan, your advisor might be able to tell you what is important to him, which of the points should maybe get more attention or less.