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The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.

Let M n g be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n g . In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n > 2 − 2g.

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