I find myself particularly drawn to the subjects of analysis and linear algebra. Although I am currently a rising Sophomore and unsure about the specific field I'd like to pursue for my graduate studies, I am eager to enhance my profile by gaining research experience through REUs (Research Experiences for Undergraduates). Additionally, I see this as an opportunity to determine whether research is genuinely a passion of mine or not.
But the common understanding seems to be that research in analysis-related fields is inaccessible to people like me without background knowledge of at least measure theory and Lp Spaces which I guess would take me at least 1.5 to 2 years to get by self-study which would be my Junior year summer time. Till then I am not sure how should I get involved enough in this field to make myself capable and indulged in the research experience. Research in discrete math seems a viable option till then (but extremely tough to get) but that's not what I would be aiming for in grad school. I am in NYC and discrete math seems to be popular here with probability.
Also an important point, I am an international student so REUs are very limited and I have part-time work to do, so can't fast-track and spend my all free time accelerating the learning process.
So my question now is:
Are there projects in analysis-related fields where a basic first course in real analysis and Linear algebra would be good enough to get started for my Sophomore summer?
My Real analysis study from Zorich Mathematical Analysis 1 should get me to Chapter 7/8 by March 2024. My linear Algebra is from Titu Andreescu's book (till Chapter 7/8 by March 2024) and lectures online.
Is accessible research topics like those from Combinatorics/ Graph theory the only research experience I can think of? (keeping in mind that even these are extremely hard for me to get)
If so would it be wise to dedicate some of my time here to be prepared enough to apply for the projects? I have liked the idea of Algebraic Graph theory but yeah I only have an idea about it not even a preview of the subject.
My mind is toying with a rather silly notion: to prioritize a minor increase in my research experience chance over dedicating time to real analysis self-study. Keeping in mind that real analysis is crucial to learn, not only due to my personal interest but also because it is a fundamental requirement for almost every graduate admission, demanding proficiency in this subject.
Also, an important sub-question, do people usually do REU in a field say A, and go on to do their graduate school in say field B? How does graduate admission look at this fact?
TLDR: Confused and just overthinking due to spare time.