I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev Embedding Theorems (more functional analysis) and one on variational elliptic problems in PDEs (which as I understand will focus more on proving the existence and uniqueness of solutions). I'm already studying Sobolev Spaces and its properties and the direction I take after this will differ based on what I choose. Unfortunately, both interest me equally which is why this continues to be a dificult decision to make.
So, I want to ask what writing or doing research on these topics are like? Is there a topic that's more important to learn as early as now for a student wanting to pursue graduate studies?
Thank you so much!