**Motivation**

Data is commonly represented as an unordered sequence of points in the Euclidean space . The global `shape’ of the data may provide important information about the underlying phenomena of the data.

For data points in , determining the global structure is not difficult, but for data in higher dimensions, a planar projection can be hard to decipher.

From point cloud data to simplicial complexes

To convert a collection of points in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance ). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:

Given a set of points in Euclidean space , the **C****ech complex** (also known as the nerve), , is the abstract simplicial complex where a set of vertices spans a -simplex whenever the corresponding closed -ball neighborhoods have nonempty intersection.

Given a set of points in Euclidean space , the **Vietoris-Rips complex**, , is the abstract simplicial complex where a set of vertices spans a -simplex whenever the distance between any pair of points in is at most .

Top left: A fixed set of points. Top right: Closed balls of radius centered at the points. Bottom left: Cech complex has the homotopy type of the cover () Bottom right: Vietoris-Rips complex has a different homotopy type (). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.

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You can make the math symbols bigger than the text font for easy reading. eg.

“&s=3” added after the last } tells latex to make the symbol lambda bigger at size 3 (which is slightly bigger than the text font, so it stands out).

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Sorry, latex should be:

““.

For color, you can append behind “&fg=aa0000 ” (brown)

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