Dr Vanessa Robins & Dr Katharine Turner, Australian National University
Topological Data Analysis (TDA) is an interdisciplinary field combining methods from algebraic topology, statistics and computational algorithms. It quantifies the shape of data over a full range of length scales and, most importantly, captures how that shape changes as the length scale parameter is varied. Applications are diverse with examples including the quantification of bone morphology and porous materials, the connectivity structure of the brain, and time series analysis.
This course will cover the relevant background from algebraic topology, provide a detailed overview of persistent homology – the main tool in TDA, and various approaches to summarising the information provided by persistent homology. Since any data analysis must consider the effect of randomness and of noise we also will study statistical aspects in TDA including stability, correlation, and statistical significance tests. The tutorials and assessment will include working with established software packages to analyse example data.
- Introduction to homology
- Filtrations and Persistent homology
- Essential algorithms in TDA
- Summaries of persistent homology information
- Statistical aspects of TDA
- Further techniques: Discrete Morse theory, the Reeb graph and Mapper
- Example Applications
- Linear algebra
- Second year abstract algebra
- Recommended to have some familiarity coding with packages such as Python, R or Matlab.
- Assignment 40%
- Final exam 60%
- “Computational Topology” by Edelsbrunner and Harer
- “Elementary Applied Topology” by Robert Ghrist.
Available via online pdf at https://www.math.upenn.edu/~ghrist/notes.html
Dr Vanessa Robins, Australian National University
Dr Vanessa Robins holds an ARC Future Fellowship in Applied Mathematics within the Research School of Physics and Engineering at ANU. Her research is interdisciplinary and draws on mathematics, physics, chemistry and computer science. Specifically, she studies the mathematical description of shape via geometry and topology, practical methods for computing such information from data, and applications of these concepts in physical, chemical and biological settings. Her doctoral work at the University of Colorado, Boulder (2000) produced one of the earliest papers on persistent homology, helping to establish the field of Topological Data Analysis. She subsequently devised mathematically rigorous algorithms for computing topological information from digital images, such as those produced by ANU’s x-ray micro-CTLab. She has contributed to the enumeration of crystalline frameworks and their potential modes of entanglement via geometric group theory and combinatorial tiling theory of curved surfaces. These investigations of theoretical structures have led to new models for materials including metal-organic frameworks, light-weight rigid isotropic microstructures, and insights into the way keratin filaments are arranged in mammalian skin. Dr Robins has lectured in the Mathematics and Physics undergraduate programs, supervised many undergraduate research students, five doctoral students at ANU, and four at overseas institutions. Her outreach activities include working with five artists in residence.
Dr Katherine Turner, Australian National University
Katharine Turner is a Lecturer at the Mathematical Sciences Institute at the Australian National University. She previously worked at EPFL, Switzerland and did her PhD at the University of Chicago.
Katharine works in the interdisciplinary field of Topological Data Analysis. She is particularly concerned in the theoretical development and application of topological summaries to a variety of applications in a statistically rigorous manner.