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I'm a sophomore undergraduate majoring in mathematics, and I'm pretty sure about eventually applying to graduate school in pure math. However, I haven't done much other than coursework so far, and I'm trying to plan a productive summer (learning something new, perhaps interacting with professors). By the time summer begins, I will have completed all basic math courses such as Analysis, Abstract Algebra, Linear Algebra, Set Theory, First Order Logic, etc. Apart from this, I've spent some time self-studying advanced topics such as Graph Theory, for example.

I think the above background is sufficient for me to at least learn about advanced math topics, if not do research at this stage. By advanced, I mean anything that would have at least one of the above topics as prerequisites, for example, Category Theory or Algebraic Geometry.


What do summer projects in math look like? I believe, for the most part, these would just be directed reading projects (which is great too). How do I find such projects?

Big Question: Would it be a good idea to cold email professors (ones who I've had no prior interaction with, but I really want to work with them at some point) and ask if they'd be willing to advise me on a reading project or such? I'm thinking of approaching professors at institutions different from mine - in particular, places where I'd like to go to grad school too. Would this be a good idea? I'm aware that professors are busy and they're not likely to respond to undergrad students from other institutions easily, but seems worth trying.

In addition, would it also be a good idea to reach out to postdocs or senior PhD students in math? I guess they could be good mentors for reading projects, too.


Lastly, if anyone has any other ideas for spending a productive summer in pure mathematics, do let me know! (apart from summer schools and REUs, I guess. Most of them are limited to US citizens/residents, making them useless for me).

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Few professors will have any incentive to work with you. They are either busy in summer months or unwinding from otherwise intense efforts. If you have a personal connection to someone it might be different, but "cold contacts" probably will fail, even not getting any response.

But there are a few things you could try.

First, it may be that your current instructors could guide you if you don't put much burden on them.

Next, they might have maintained contact with their own teachers and can introduce and recommend you to a distant university professor.

Third, you can find the course requirements online for some topic of interest to you and get a copy of the textbook. Get the syllabus if available and follow it. This will give you access to readings and possibly exercises that you can do. Do a lot of them. Getting feedback on your results will be an issue, but maybe someone local can help you.

Finally, you might consider forming a study group if you can find others who are interested and work together to learn and give each other feedback. A local professor might help you arrange such a thing and even provide minimal guidance.

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