The Persistent Homology Pipeline: Shapes, Computations, and Applications

a possibly large and possibly noisy data collection.
A commonly desired goal is to obtain global summaries of the data
which help to understand the data on a high-level and simplify
the exploration process.
Interpreting data as geometric input, such summaries are provided
by topological invariants, for instance the homology of spaces,
which leads to the field of topological data analysis. The last 20 years
have witnessed a boost of this research area,
mostly due to the development of persistent homology.
This is a theory to
make topological invariants robust with respect to noise and yielding
a topological multi-scale summary of data.

The success of persistent homology has posed the challenge
of computing persistence on large data sets. Typical questions
in this context are: How can we efficiently build combinatorial cell complexes
out of point cloud data? How can we compute the persistence summaries
of very large cell complexes in a scalable way? Finally, how does the computed
summary lead us to new insights into the considered application?
My talk will introduce the theory of persistent homology on an informal
level, discuss some recent algorithmic advances and
survey some application areas of topological data analysis.