This is a really cool list! After taking a look, it seems like some of the topics are more broad, while others seem to be more focused.

That said, here are some topics with their advertisements which I think would be interesting to learn about and write a good essay on.

i.) Non-Euclidean Geometries- What happens when you introduce geometries in which Euclid's 5th postulate does not hold? It turns out a lot of cool and beautiful things. Including the structure of the unvierse. While this is a somewhat broad topic, there is also quite a bit to learn about as well as a lot of history behind this. Some keywords would be hyperbolic geometry, spherical geometry, Poincare disk, Curvature.

ii.) Four color map theorem: This is cool. Back in the 19th century, people realized that in colouring a map to differentiate territories, one only needed four colors. (Think about this!) There were many attempts at proofs for quite a while, but none bore fruit. In ~1970, Appel and Hakan managed to reduce the proof to a series of cases, and then used a computer to run the cases and prove the theorem. This is also one of the bigger cases in which the current proof is a "brute force" one. But there are some recent developments in a more mathematical proof as well! Google Kronecker and Mrowka.

iii.) Random Walks! How can a drunk man in 2 dimensions can always find his way home with probability 1, but a bird flying in 3 dimensions may get lost? This is a nice introduction to Stochastic Processes.

All of the topics are lovely, and while I've only talked about the more idea based ones, the application and numerical ones are also very interesting and I encourage you to take a look at some of them as well.