General course info
- Neža Mramor Kosta (firstname.lastname@example.org)
- Aleksandra Franc (email@example.com)
- Gregor Jerše (firstname.lastname@example.org)
Introduction. Basic definitions and concepts: topologies, metrics, continuous maps, homeomorphisms.
Plane triangulations, Voronoi diagram, Delaunay triangulation.
Properties of the Delauney triangulation. Geometric simplicial complexes, examples
Friday lectures (instead of next Monday, March 12th when there will be a lab session instead of lectures):
Abstract simplicial complexes, examples, links, stars, triangulated surfaces and manifolds. Orientation. Simplicial complexes on data sets: the Vietoris-Rips construction.
Triangulated manifolds, orientation. Classification of surfaces.
Simplicial complexes on data sets:
Levenshtein distance calculator:
Homotopy of maps, homotopy equivalence of sets, contractible sets
No lectures this Monday, happy Easter!
Examples of nerve complexes: Čech, Delaunay, alpha-shapes
Towards homology groups: groups, homomphisms, structure theorem for finitely generated abelian groups, quotien groups, chains
A nice explanation of the algorithm to compute homology:
You can also use Dionysus to compute homology (and much more):
Wolfram Demonstration of computing simplicial homology of an alpha complex:
Homology with Sage:
Homology (and more) with CHomP:
Morse homology, algorithm for computing Morse homology, cancelling critical cells
Discrete Morse Theory:
Morse functions for data analysis, algorithm for extending sampled values on vertices to a discrete Morse function on the complex
Sublevel complex filtration, regular events and elementary collapses, critical events
Persistence: barcodes, persistence diagrams and persistent homology groups
Stability of persistence diagrams