I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.

I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2\times 30$ hours of lectures.

Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!

I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!