We use persistent homology to build a quantitative understanding of large complex systems that are driven far-fromequilibrium; in particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh-Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

The Euler calculus -an integral calculus based on Euler characteristic as a valuation on constructible functions -is shown to be an incisive tool for answering questions about injectivity and invertibility of recent transforms based on persistent homology for shape characterization.

We present a generalization of the induced matching theorem of [1] and use it to prove a generalization of the algebraic stability theorem for R-indexed pointwise finitedimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance. 1 for choices of t ∈ [0, ∞). The collection {C(f, t)} t∈R is called the sublevel set filtration of X induced by f . Superlevel sets and superlevel set filtrations are defined similarly by considering the sets {x ∈ X : t ≤ f (x)} for every t ∈ R.Alternatively, assume that X is a topological domain and f : X → R is a scalar value of a nonlinear physical model, e.g. the magnitude of vorticity or temperature field of a fluid, the chemical density in a reaction diffusion system, the magnitude of forces between particles in a granular system, etc. Patterns produced by these systems are often associated with sublevel or superlevel sets of f . In fact, the direct motivation for this work is to justify claims made in [16] concerning the time-evolution of patterns in convection models.These examples are meant to motivate our interest in studying the geometry of the sets C(f, t). Homology provides a coarse but computable representation of this geometry. In particular, for each t ∈ R, there is an assigned graded vector space, the inclusion maps induce the following linear maps at the level of homology:This homological information can be abstracted as follows.Definition 1.1. A persistence module is a collection of vector spaces indexed by the real numbers, {V t } t∈R , and linear maps {ϕ V (s, t) : V s → V t } s≤t∈R satisfying the following conditions:We write (V, ϕ V ) to denote the collection of vector spaces and compatible linear maps, and will sometimes just write V for the full persistence module when the maps are clear. We say that V is a pointwise finite dimensional (PFD) persistence module when every V t is finite-dimensional.As is described in Sections 2.1 and 4, a PFD persistence module gives rise to a persistence diagram, which is a set of points inGiven a PFD persistence module (V, ϕ V ), we denote its associated persistence diagram by PD(V ). Observe that we have outlined a procedure by which the sublevel sets of a scalar field f produce a persistence diagram PD. Returning to our examples, in the first case, it is reasonable to assume that the actual available data is X ′ ⊆ X ⊆ X, as opposed to X , which represents the true set of objects upon which the clustering is to be based. In this case, collecting experimental or numerical data results in f ′ : X → R, an approximation of the actual function of interest, f . Recent computational developments have led to the routine computation of PD ′ , the persistence diagram associated with X ′ or f ′ . Thus, the natural question is this: how is PD ′ , the computed persistence diagram, related to PD, the persistence diagr...

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