Is it adequate to expect to accomplish some amount of original work in a masters thesis?
closed as not a real question by Pete L. Clark, Arturo Magidin, Jonas Meyer, Sivaram Ambikasaran, Willie Wong♦ Feb 19 '11 at 9:27
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The answer depends on the standards of your institute, but even in good institutes, it is often acceptable to just present a summary of a difficult paper, with all details filled in. While certainly not original work, it can be difficult for the student since in math often results are presented in a very cryptic way, as part of the conventions of the subject (this is very different in, say, computer science).
It is always better if you do some original research, but you may not be mature enough for that, or versed enough in the subject you're hoping to work on. Original work comes later, during your PhD, which is significantly longer, and having an appreciably higher starting point.
Your question is abstract and needs to be made concrete. Much like finding a husband or wife, you don't need to know how to charm every person in the world, just one. In this case, you only need to satisfy one relatively small group of people at a particular institution.
Identify the people who will directly need to approve your work. For instance, it might depend on a particular adviser. Alternatively, there may also be a committee. As people have said, there are no real rules here. There is no central authority that forces every mathematics department in the world to do the same thing; and so, as far as I know, they don't. After you have figured out the main players, figure out their relative importance. It may turn out that if you adviser approves, then everybody else will rubber stamp it. This may be the case even if the rules on paper say otherwise. Bureaucratic institutions can be like that and universities are quite bureaucratic. It may turn out that you need to have some committee's approval. There may be some people who will not bother to read it. There may be others who will nitpick everything and make you rewrite everything a dozen times. The best way to avoid nasty surprises is to get some intel on the people involved. The easiest way to do this is to just go and ask directly, "Prof. X, I am planning to do Y for my thesis. Would that be what you are looking for?". Later you might have to say, "Prof. X, I tried to do Y, but I only managed to do Z. Is that enough?" If you keep him informed, then it shouldn't be a surprise to you when you turn in the final copy whether he will approve it or not. You can also get information from people who have been through the same process, other graduate students.
This all involves a lot of chatting and people skills and you may not want to do this. However, the alternative is you are wasting time and money. It's okay to approach this in a professional way. I think most professors will respond professionally and provide you with the information you need to make informed decisions.
If you don't have an adviser yet, try to schedule a few meetings and try to specify as much as possible what you might be working on. Pick the project that has the best chance of allowing you to complete everything adequately and on time.