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I am currently a sophomore majoring in mathematics. I am interested in pursuing a research career in my future. In order to have a taste of mathematical research, I contacted a professor who could lead me if I join a university-organised summer research program. He assigned me a number theory problem that he believes would suit my background. However, as I read the paper he assigned me, I felt stuck almost everywhere. I might at most be able to explain a rough outline of the paper (i.e. how each part of the paper contributed to the eventual proof of the central theorem). The summer research program application deadline is in less than 3 weeks, and I have no idea if the professor will be satisfied with my progress and agree to supervise me during the summer research program.

Nevertheless this is not the whole picture of my problem. I feel like I am more interested in devoting myself to analysis/geometry, and I chose the number theory professor because I think my background knowledge in both analysis and geometry is be too weak, and number theory seems appropriate for me to have a taste on math research as a sophomore. Should I keep attempting for number theory (perhaps not having an opportunity to conduct research over the summer), or should I concentrate on reading books/taking more courses on analysis and geometry, and contact a professor on these two areas during my junior year? I am worried that not starting research until junior year might be too late if I want to apply to a graduate school.

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As a second year student, I don't think there is a downside to any option. Personally (math analysis) I find number theory daunting, but there is more action there currently I think.

My suggestion would be, assuming it is possible, is to have a face to face discussion with the professor about the paper, indicating where you are having difficulty. There is nothing wrong with that.

Don't neglect other opportunities, but don't consider yourself to be a failure for not understanding something that might be quite deep. You've got to start somewhere. You lose little by following up on this opportunity. It will, perhaps, give you some insight even if you don't make much progress with it.

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"Undergrad research" in mathematics is a tricky thing. In the U.S., especially, undergrad math majors don't really take very many math courses, in comparison to other countries, due to the different university ideas. Even so, no, a year or two of basic mathematics is not at all adequate to begin "research" on topics that far more experienced/scholarly people have attempted and (presumably) failed. How could this be conceivable?

(Yes, I know, in some engineering or CompSci situations, being a worker in a lab is "research", but there's rarely an analogue in mathematics.)

There are some typical traps for people wanting to "do research" in math, without much background: graph theory, and elementary number theory. The latter is especially insidious, because people've been looking at it for centuries. Graph theory less so. And, hey, turns out that lots of other parts of math (complex analysis, abstract algebra, functional analysis, Lie theory, algebraic geometry, ...) are essential to understand, much less do, modern number theory.

So, without knowing what you've been looking at, it's entirely plausible that it's a reasonable article, but uses some other math that you don't know.

And, to answer another of your questions: yes, keep studying! :)

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