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Module MAU34602: Applied Algebraic Topology

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2020
Contact Hours
10 weeks, 3 lectures including tutorials per week
 
Lecturer
Prof. Colm Ó Dúnlaing
Learning Outcomes
On successful completion of this module students
  • will be familiar with the practical aspects of homology theory;
  • will be familiar with abstract and concrete simplicial complexes;
  • will be able to calculate some homology groups and persistent homology groups, and have experience in some aspects of writing code for these calculations;
  • will have encountered some other interesting applications of (algebraic) topology.
Module Content
  • Simplicial complexes, abstract and geometric.
  • Collapsing and graph evasivenes.
  • Other kinds of complex: Čech, convex set complexes, Vietoris-Rips, Delaunay, alpha.
  • Homology groups. Motion of objects in contact (Hopcroft and Wilfong). Concrete methods of calculating homology groups.
  • Persistent homology groups and their calculation.
  • Some graph-theoretic and related algorithms will be introduced and used where needed.
Module Prerequisite
MA3428 (Algebraic Topology II), knowledge of C or C++ programming.
Assessment Detail
This module will be examined in a 2-hour examination in Trinity term. There will be fortnightly quizzes and some programming assignments. The overall mark will be 80% exam, 10% quizzes, and 10% programming. Re-assessments if required will consist of 100% exam.
Text
Edelsbrunner and Harer. Computational Topology: an Introduction. The chapters on graphs, complexes, homology, Morse functions, and persistence, will be used.