Prerequisites

  •  Rudimentary knowledge of programming in Python.
  • An undergraduate course in topology (e.g. chapter 1-3 of J.R. Munkres: Topology should be familiar).
  • The student should be familiar the construction of free vector spaces and quotient vector spaces, representation of linear maps by matrices, and matrix operations such as multiplication and row/column operations. We will mostly be concerned with finite-dimensional vector spaces over the field of two elements (and occasionally other prime fields).

Aim of the course

Topological Data Analysis (TDA) is a recent approach to data analysis in which tools from (algebraic) topology are applied to gain a qualitative understanding of data - or in other words - to infer properties of the shape of the data. The ‘’simplest’’ of the topological invariants considered is the number of connected components of a data set. Translated into the language of traditional data analysis this would correspond to the task of clustering the data, i.e. the process of grouping data points together such that points in the same cluster are comparatively closer to each other than pairs formed from different clusters. We will see how tools from (algebraic) topology can be used to detect other types of non-linear structure in data. Applications to materials science, neuroscience and sensor networks will be discussed.

The aim of this course is for the students to learn about the algorithms and mathematics underlying the core tools of TDA, as well as to explore the applicability of TDA to the sciences.

Topics covered include:

  • Simplicial theory: fundamentals of simplicial complexes, simplicial (co-)homology (incl. algorithms), simplicial complexes from data (Cech, Rips, Delauney, Alpha), Nerve Lemma, sensor networks.
  • Persistent (co-)homology: algorithms, algebraic foundations, stability, homological inference, interleavings, circular coordinates, kernel methods, applications.
  • Multiparameter persistent homology.
  • Clustering: ToMATo, Kleinberg’s theorem, clustering in the language of category theory, multi-parameter clustering.
  • Aspects of smooth (computational) Morse theory: Morse theory for surfaces, Morse-Smale complex, Reeb graphs and Mapper.
  • Discrete Morse theory.

Elements of algebraic topology and category theory such as homotopy equivalence and functoriality will be introduced when needed.

Lecturer

Magnus Botnan (VU)