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Matthew L. WrightMatthew L. Wright Institute for Mathematics and its ApplicationsInstitute for Mathematics and its Applications University of MinnesotaUniversity of Minnesota

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4 For a polyhedron, ? – 6+ 4= 2 verticesedgesfaces (# vertices) – (# edges) + (# faces) =

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For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

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For a polyhedron, ? (# vertices) – (# edges) + (# faces) = 8– 12+ 6= 2 verticesedgesfaces

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For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

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For a polyhedron, ? (# vertices) – (# edges) + (# faces) = The Euler characteristic of a polyhedron is 2.

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Euler Characteristic, more generally: Finite set Finite graph, no loops Finite graph, with loops Compact planar region = 8= 8 – 5 = 3= 8 – 6 = 2= 8 – 12 + 3 = -1 (number of connected components) – (number of holes)

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Nota Bene Euler Characteristic is independent of decomposition: = 1 …but is sensitive to open/closed sets: = 0 = 1 = -1

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Key Property This property is called additivity, or the inclusion-exclusion principle.

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Integral with respect to Euler Characteristic

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1 2 3

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Preferred in practice! Proof: telescoping sums Now integrate.

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2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

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2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

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2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4 compute: There are 7 sets! sum 2 1 3 4 5 6 7

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Proof: Therefore,

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Application: Euler integration is useful for enumeration problems (Baryshnikov and Ghrist).

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Can we count by sensors? I count one. sensors network I count none.

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Can we count by sensors? ??? ? ? ?

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How can we compute Euler integrals over networks?

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Computation:

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Example: Start with a grayscale digital image.

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246 635 52 25 164 236 31 56 42155

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246 635 52 25 164 236 31 56 42155

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246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

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246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

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246 635 52 25 164 236 31 56 42155 threshold Euler characteristic Euler characteristic graph

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Richardson and Werman used the Euler characteristic graph to classify 3D objects. a catcurvature function on the cat curvature above threshold 0.05

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Richardson and Werman used the Euler characteristic graph to classify 3D objects.

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The Euler characteristic curve helps distinguish between similar artifacts from two sites.

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Euler characteristic is a simple topological invariant. Homology is a much more sophisticated topological invariant. Betti numbers Connection:

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Persistent Homology e.g. components, holes, graph structure e.g. set of discrete points, with a metric Persistent homology is an algebraic method for discerning topological features of data.

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Example: What topological features does the following data exhibit? Problem: Discrete points have trivial topology.

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Idea: Connect nearby points.

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Idea: Connect nearby points, build a simplicial complex. 3. Fill in complete simplices.

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…then we detect noise.

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A collection of bars is a barcode.

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?

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