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I am thinking of doing some research (Real Analysis preferably) under a professor this year and I need ideas for interesting topics that might be accessible to an undergraduate.

Does anyone have any starting points for me?

I have taken Calculus, Multivariable Calculus, Differential Equations, Honors Linear Algebra I, Honors Abstract Algebra I, Honors Real Analysis I, as well as two proof-writing classes. You may assume that I am willing to learn any accessible material to fill in gaps in my knowledge. I have a preference for Analysis, though I am open to suggestions in other areas.

My initial idea was (Compact = Closed + Bounded + (?)), but in trying to write my proposal for the research (only two pages!) I found it hard to put into words exactly how I was going to go about the research, or how I would focus my efforts.

Please if you have any suggestions for either the topic of research, or where I could go to find the topic (obviously I am taking input from my professors already), let me know.

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Shouldn't the professor you are planning to work with be the source of the ideas, rather than yourself? – Arturo Magidin Nov 25 '11 at 22:16
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These responses are not helpful, and they also encourage others to give responses which also do not address my question. As stated before, I am taking input from my professors, but they are not infinite sources of research topics, and I was looking for a fresh perspective. – nullUser Nov 25 '11 at 22:24
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Kb100: those responses are helpful, because the reaction of most people with experience will be that an undergraduate interested in doing a research project with a professor shouldn't be the one picking the topic. For example, your initial idea isn't really suitable for research because in metric spaces "compact = complete + totally bounded" has been known for a long time. Maybe you might want to consider doing a reading course on some substantial advanced topics as an alternative. – KCd Nov 25 '11 at 22:31
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@Kb100: You are essentially asking people who are not directly familiar with your situation, and who will not be involved with the project, to suggest things you might do. Any input you get from here should be worth $\epsilon$ (for a very small $\epsilon$) compared to the input you get from your professors. It also seems that your problem lay rather with an inability to write up a proposal, rather than a choice of topics. That's something you could ask your professor for help with; writing proposals is hard, and is one of the things you could be learning with the project. – Arturo Magidin Nov 25 '11 at 22:34
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I agree that the usual order of business for under/graduate research is for the faculty member directing the research to largely or entirely suggest the topic of research. But that's not strictly inevitable, and I think that the collective knowledge of undergraduate real analysis assembled here compares favorably to that of most individual mathematicians. I am tempted to suggest some topics for undergraduate research myself and see what kind of feedback I get from others here. But, of course, this requires some thought. If the question sticks around, I will probably do this... – Pete L. Clark Nov 25 '11 at 23:49
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closed as not constructive by Austin Mohr, Lord_Farin, Jim, Julian Kuelshammer, vonbrand May 14 at 8:18

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