The Australian Mathematical Sciences Institute 2018 Summer School in the Mathematical Sciences at Monash University has just finished. I took the topological data analysis and low-dimensional topology courses.
Topological data analysis
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)
- 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