Ideas for Undergrad Research Project

Question

I'm in my second year of my maths undergrad course and hoping to do a 6 week research project this summer.

I'm interested in doing a project wherein I apply a maths topic, (i.e Manifolds, Metric spaces and topology, Groups, Rings etc. preferably the more abstract topics) which I've studied and use it in algorithms. I've come across a few topics which seem interesting but my knowledge of them is limited at the moment. I have decent programming ability not a lot of experience but hope to learn more. I know Python, C and R to some extent

1. Manifold learning: This seems very interesting to me the use of manifolds to tackle high-dimensional data sets. I have only just encountered Manifolds this term and I am an unsure I have the ability/knowledge to take on a project of some sort in this.

so my questions with regards Manifold learning are;

What is required for Manifold learning (or other uses of manifolds in programming ,if there are any) on the maths side, statistics side and CS side? if there are other topics matching up programming and Manifolds what are they and would they be feasible at my level?

2. Computational Topology: From what I have read so far is that algebraic topology is required for this field, which I haven't done yet. My experience with topology is a semester in Metric and topological spaces.

So my question for Computational toplogy is;

Is there any part of this field I could do a project in with my current knowledge?

3. Groups, Rings, Fields: I haven't really looked into the uses of these topics as much as the first two. I know encryption uses these topics but other than what crossover between these topics and algorithms are there?

Lastly, if anyone has any suggestions of further reading or possible projects for these subjects or related to these subjects or anything you think I might find interesting they would be much appreciated. I will most likely work with a supervisor to come up with my actual project so I really just need some guidance so I can suggest to them what area I am looking at doing a project in.

Thanks in advance :)

P.s Would you advise getting a supervisor from the maths department or CS department?