Weekly outline

  • General course info

    Lecturers: 

    • Neža Mramor Kosta (neza.mramor@fri.uni-lj.si)
    • Žiga Virk (ziga.virk@fri.uni-lj.si)
    Assistants: 
    • Aleksandra Franc (aleksandra.franc@fri.uni-lj.si)
    • Gregor Jerše (gregor.jerse@fri.uni-lj.si)

  • 18 February - 24 February

    Introduction. Basic definitions and concepts: metrics, continuous maps, homeomorphisms. 

    • 25 February - 3 March

      Triangulations in the plane, Voronoi diagram, Delaunay triangulation.

      • 4 March - 10 March

        Geometric simplicial complexes, Abstract simplicial complexes, Euler characteristic

        • 11 March - 17 March

          Triangulated manifolds, orientation. Classification of surfaces.
          • 18 March - 24 March

            Simplicial complexes on data sets:

            • Vietoris Rips complex
            • The nerve construction


            Levenshtein distance calculator:

            https://planetcalc.com/1721/

            • 25 March - 31 March

              Homotopy of maps, homotopy equivalence of sets, contractible sets

              Nerve theorem


              • 1 April - 7 April

                Homology groups

                • 8 April - 14 April

                  Computing homology

                  • 15 April - 21 April

                    Persistent homology


                    A nice explanation of the algorithm to compute homology:

                    https://jeremykun.com/2013/04/10/computing-homology/

                    You can also use Dionysus to compute homology (and much more):

                    http://www.mrzv.org/software/dionysus/

                    http://www.mrzv.org/software/dionysus/python/overview.html#python-bindings

                    Wolfram Demonstration of computing simplicial homology of an alpha complex:

                    http://demonstrations.wolfram.com/SimplicialHomologyOfTheAlphaComplex/

                    Homology with Sage:

                    http://doc.sagemath.org/html/en/reference/homology/index.html

                    Homology (and more) with CHomP:

                    http://chomp.rutgers.edu/Software.html


                    • 22 April - 28 April

                      Persistent homology: bar codes, persistence diagrams


                      • 29 April - 5 May

                        Enjoy your holidays!


                        • 6 May - 12 May

                          Sublevel set persistence, persistence on data sets, bottleneck distance


                          • 13 May - 19 May

                            Stability of persistence diagrams

                            Morse theory: discrete vector fields, discrete Morse functions, sublevel omplex filtration: critical and regular events

                            • 20 May - 26 May

                              Discrete Morse Theory: 

                              Morse functions for data analysis, algorithm for extending sampled values on vertices to a discrete Morse function on the complex

                              Software for computing persistent homology

                              • 27 May - 2 June

                                Friday: project presentations