Persistence is (not) Morse Theory

I will survey the relationship between Morse theory and persistent homology from different aspects: historical connections, discrete Morse theory of geometric filtrations, simplification by Morse cancelation of persistence pairs, algebraic Morse theory, and persistence computation. I will illustrate how the conceptual differences between persistence and Morse theory provide complementary viewpoints, and how an understanding of their interplay can lead to drastic improvements in computational methods.