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I’m an undergraduate math major and am currently a rising sophomore. In my first year, I had a fair bit of experience with real analysis and linear algebra (but nothing fancy). I have some free time this summer, and so I’m thinking of working on a project (preferably a short paper) in math. I’m also in touch with a math professor who’s willing to help me out with this. Considering my limited experience, do people have any recommendations on areas I can focus on? Of course, I don’t hope to produce anything original but I definitely want to work on something paper-worthy which is in my reach, even if it requires a lot of work (I have time).

I've been told that I should focus on writing an expository paper/survey rather than an actual research paper, so I would love to hear your recommendations on what topics I can/should look into for that. I'm not looking to be spoonfed ideas, I just want to hear about what the math community finds interesting for an undergrad at my level - something challenging (beyond what's done in class) but still doable. I'm not specifically looking for research questions per se, I'm also fine with interesting areas related to my experience that I can learn about, read papers on, and write about.

Some topics I'm reading up on and would be excited to delve further into: functional analysis, measure theory, Fourier analysis

PS - I'm also perfectly happy with other suggestions regarding what math-related thing I can do with my time this summer.

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