Topological data analysis is a field which uses ideas and techniques from topology to analyse and characterise data sets. This course covered a lot of ground quite quickly:
We can approximate a topological space by simplical complexes (which is closely related to the familiar triangulation used
Using simplical complexes to represent topological spaces and computing their homology.
Simplical complexes and computing them from data
The homology of simplical complexes
Persistent homology, which is the main tool of TDA
Comparing persistence diagrams (the space of persistence diagrams)
Statistical analysis of persistence diagrams (monte carlo simulation)
Some applications
Functional summaries of persistence diagrams (rank, landscape, persistence image)
Functional principal component analysis (FPCA)
Union-Find for connected components
Kruskal’s algorithm for MST
Smith Normal Form for computing the boundary matrices
An incremental algorithm for computing Betti numbers
An algorithm to compute persistent homology by pairing simplicies
Morse theory for smooth manifolds
Discrete Morse theory
A reception
A closing dinner.
A diversity session and panel discussion.
Morning tea every week day.
BBQ lunch each Wednesday.
A lunchtime lecture each Tuesday.
A maths-related movie night on Thursday night.
An excursion to the Philip Island and the penguin parade.
An excursion to the Yarra valley to visit a winery or two.
A tour of the Monash University Wind Tunnel Facility