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MA941 - Topology of Data

Lecturer: Martin Lotz

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Prerequisites: linear algebra, familiarity with topology

Content :

Topological Data Analysis (TDA) is an approach to data analysis based on techniques from algebraic topology. Topology is the study of properties of sets that are invariant under continuous deformations; it is concerned with concepts such as ``nearness'', ``neighbourhood'', and ``convergence''. Nowadays, topological ideas are an indispensable part of many fields of mathematics, ranging from number theory to partial differential equations. Algebraic topology, in particular, aims to understand topological properties of spaces through algebraic invariants. The premise of topological data analysis is that data there is an underlying topological structure to data. Familiar examples include clustering, where the aim is to subdivide data into different clusters, or ``connected components'', and connectivity in networks. In this module we introduce persistent homology, a powerful method for studying the topology of data. We discuss the theoretical foundations, as well as computational and algorithmic aspects and various applications. While the course is mainly theoretical in nature, you are encouraged to experiment using a range of available software and applications. The lecture material will be available as video recording and slides, and exercises will be published semi-regularly.

Intended Learning Outcomes

Upon completion of this module you should be able to:

  • understand how topological information can be extracted from discrete data;
  • use persistent homology to compute persistence diagrams and barcodes;
  • explain the different parts of the persistent homology pipeline and the computational challenges involved;
  • evaluate the stability and robustness of persistent homology computations;
  • summarize different approaches to the topology of data and discuss applications

 

References:

  1. Steve Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS 2015
  2. Herbert Edelsbrunner and John Harer. Computational Topology, An Introduction. AMS 2010
  3. Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod & Heather A Harrington. A roadmap for the computation of persistent homology. 2017

More specialised sources and papers will be made available in time.

See the Moodle page for lecture material.