In the summer of 2020, I participated in the University of Chicago Math Research Experience for Undergraduates. Over the course of this program, I wrote an introductory paper on Gromov's h-principle, which is a very interesting topic within homotopy theory, a subfield of topology. You can find the paper hosted on the REU website here.
A highlight of the paper is proving the sphere eversion, i.e. the fact that you can smoothly turn a sphere inside out if you allow self-intersection. There's a great video on this well-known among math nerds: https://www.youtube.com/watch?v=wO61D9x6lNY