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In the last 15 years, persistent homology emerged as a particularly active
topic within the young field of computational topology. Persistence tracks
the evolution of homology classes and quantifies their longevity. By
encoding physical phenomena as real-valued functions, one can use
persistence to identify their significant features. This talk will
introduce persistence, discussing the settings in which it is effective as
well as the methods it employs. It will also describe two extensions to
persistent homology, zigzag persistent homology and well groups, and how
all three relate to each other, through a Mayer--Vietoris pyramid, when the
input data is a real-valued function.