Before I ask my question, this is my first post on MSE, so I apologise if I have not met the standard etiquette of a post.
I have finished my first year at a UK univeristy for a maths degree. In our second year we are asked to conduct some form of research on a topic of our choice. (I don't have much idea what this entails, but I know it's not finding anything original and proving it ourselves, it's more about wider reading.)
I've very much enjoyed my first year analysis content (sequences, series, continuity, differentiability, integrability, and power series). This has all been done on the reals.
I really want to look at more analysis content, I am interested in looking at multivariable or complex analysis but we will be taught that in our 2nd year so I wouldn't be allowed to do my project on those. Most stuff I've seen outside that seems locked behind norms, measures or metrics - these topics also have the issue of being taught in the 2nd year.
I'm sure I would be able to do a topic that dives into these a little bit and don't need much from them, but I wouldn't if the bulk of the topic requires establishing a lot of results from those topics.
I've been considering looking at non-standard analysis but I don't think my understanding of logic (doing fitch-style proofs) is sufficient.
For reference, outside of analysis we have studied a bit of groups, rings and fields, some linear algebra on finite dimensional vector spaces, basic set theory and a tiny bit of number theory, and multivariable calculus (not multivariable analysis).