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I'm a third year undergrad with this summer off so would appreciate some material to look at.

I took courses in Galois theory, Topology, Complex analysis, ...

My main interest is in Analysis / topology.

I know this is a vague question, but I have begun the holiday - so am in a different country to my professors - who I won't see until the end of Summer.

edit: hopefully a question that can be solved by next summer (or semester - I have not used that term before, though I am guessing it means acedmic year).

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You can always browse over on mathoverflow.net. Are you looking for a problem you can make some progress on before the next semester? –  Dimitrije Kostic Dec 21 '11 at 23:51
    
@DimitrijeKostic I had a glance at MO though the questions asked are tough to me. Certainly, I don't know enough mathematics as of now to attempt anything worthwhile - you are correct in guessing I want to solve it by the start of next semester, I should have included that originally; thankyou for pointing it out. –  Adam Dec 21 '11 at 23:55
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It is most practical if the research has supervision, or at least guidance. Perhaps you have a good enough relationship with someone at your university to ask for suggested reading, problem? –  André Nicolas Dec 22 '11 at 0:15
    
Why take the summer off? If you want to do research, apply to an REU! –  jspecter Dec 22 '11 at 1:11

2 Answers 2

up vote 2 down vote accepted

I wrote an article which is in the last issue of the Harvard College Math Review (a math journal for undergraduates) about an open problem in set theoretic topology that might be a good project for an undergraduate. It is called The Toronto Space Problem. I also asked a question about it in Mathoverflow.com. I think there are some references there.

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Here's a link to Manuel's question: mathoverflow.net/questions/24144/… –  Grumpy Parsnip Dec 22 '11 at 0:15

You can take a look at the following books which were especially written on the open problems in Topology

Open problems in topology - J. Van Mill, George M. Reed Google Books Link

Open problems in topology II Volume 2 - Elliot Pearl Google Books link

I hope you will find some interesting problems here.

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