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Thanks for the updated list, Anthony!

On the practical side, what packages are you guys using? I'm familiar with JavaPlex for Matlab:

http://appliedtopology.github.io/javaplex/

For Mapper consider using PythonMapper by Daniel Müllner ( http://danifold.net/mapper/ ). The UI is touchy - I use it mostly via scripts only. Mapper is pretty simple to implement (just a series of well understood pieces - the magic is in the interpretation/understanding what you've done). As an example consider the kepler-mapper project ( https://github.com/MLWave/kepler-mapper ). More lines of code are used for calling out to the d3 visualization than implementing the core mapper algorithm.

For my persistent homology calculations I always use Dionysus ( http://www.mrzv.org/software/dionysus/ ). Rumor has it a much improved parallelized version will be released soon.

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