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Well, I like maths, I like studying it and applying it. But I always feared the moment when I'd actually have to do a thesis. I'm in my third year of bachelor's studies. What I was wondering about is what kind of topic should I be looking for, for my thesis? It just is hard to me to see how I could actually research and contribute something to the field as a bachelor's student, as in something not trivial and that no one with much more education has come up with before.

From merely passing courses to coming up with something of mine is a huge leap for me, and I can't see how I can do something meaningful with the limited education I have. I'm feeling a bit like in maths, before say a phd there would be nothing new for me to come up with. I am interested in cryptography, so I was given by one of my teachers a paper on elliptic cryptography that seemed interesting and a nice starting point. But now I realise I don't know where to head.

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    The usual expectation in mathematics is that one give an original exposition of known material. This means that one digests and understands on one's own terms things already known, maybe filling them out with well chosen examples, and provides a coherent expository account. Only rarely does an undergraduate math thesis contain new research; the only student I've known who wrote a thesis containing real novelties later won a Fields Medal. It is the advisor's task to help you find a suitable and well delineated topic. – Dan Fox Jan 3 '15 at 17:33
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    @DanFox, you should make your comment into an answer. – Bill Barth Jan 3 '15 at 18:44
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What is expected varies dramatically between different schools. In my experience, bachelor's projects in math can vary from trivial to nontrivial, and also from expository to original research, and the two scales are somewhat independent. I have supervised both trivial and nontrivial expository papers, as well as several undergraduate original research projects.

The concerns you have about being able to do research are common, but they are somewhat misplaced. You can almost certainly do more research than you believe you can do.

Here is an anecdote. In the 1980s, U.S. math departments began to offer "Research Opportunities for Undergraduates" (REUs), summer programs for undergrads to work on research. At the time, these were viewed very skeptically by many mathematicians. I have heard personal anecdotes that the NSF and NSA gave out a few initial grants on a purely experimental basis, not expecting to see very good results. But the projects flourished! Now there are many math REUs, and grants that allow schools to offer them are very competitive. The math community went from doubting that undergraduate research was possible to embracing it. Many math faculty are now required, or at least strongly encouraged, to mentor in undergraduate research if they expect to get tenure. See "Undergraduate Research in Mathematics Has Come of Age" from Notices of the AMS, 2014.

The key thing is to find a good project - which means finding a good advisor. Your advisor is responsible for finding a question that is accessible to you, for pointing out the background that you need to learn, and for mentoring you as you go. You still need to work on the research - we won't do it for you - but we don't expect you to operate as if you have already earned a PhD.

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As the topic is still relevant to many others, please forgive my very late reply. (I sincerely hope that the person who posted this question did well and is in no need to read my response.)

Most bachelor theses are like the "appetizer" for a specific problem. It contains enough knowledge to get a good idea about the area, but it's kept simple so that "rookies" of that area still understand the content.

The most important questions should be covered. E.g:

(i) What am I talking about? -> Solving a PDE by approximations. (ii) Why? -> To approximate the dynamics in fluids (iii)Relevance of this thesis -> Explaining the advantages and disadvantages of my approach. (iv) How? -> Mathematical theory (v) How exactly? -> Explaining my code (vi) How good is my approach? -> Calculate accuracy (vii)Conclusion

As mathematics often tends to be, I sometimes feel overwhelmed with all the needed input. But as long as the chosen topic interests you (and yes, a good supervisor is very helpful), it's quite nice to deepen one's knowledge in a specific area.

Hopefully I could help someone with my experience.

I wish everyone success and passion!

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