This expository talk will give an introduction to the concept of persistence barcodes from topological data analysis, assuming a familiarity with simplicial homology theory. We will start with a concrete definition of persistent intervals and the theory of persistent vector spaces. Afterwards, we will discuss some basic computational and optimization techniques. As time permits, we will also sketch a theoretically interesting way to parallelize the computation of persistent intervals, drawing insights from the Mayer-Vietoris spectral sequence. The appropriate references or further readings will be provided during the talk. Note that I will be giving a similar talk, but assuming less knowledge of topology, in the Student AIM Seminar this week.