Topological data analysis
Weekly outline
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Lecturer:
- Žiga Virk (ziga.virk@fri.uni-lj.si)
- Aleksandra Franc (aleksandra.franc@fri.uni-lj.si)
- Gregor Jerše (gregor.jerse@fri.uni-lj.si)
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A topologist is a person, who doesn't know a donut from a coffee cup
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Introduction. Basic definitions and concepts: metrics, continuous maps, homeomorphisms, homotopy type.
Textbook: Chapter 1
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Triangulations in the plane, Voronoi diagram, Delaunay triangulation.
Textbook: Chapter 2
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Geometric simplicial complexes, Abstract simplicial complexes, Euler characteristic
Textbook: Chapter 3
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Triangulated manifolds, orientation. Classification of surfaces.
Textbook: Chapter 4 -
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Carve a pumpkin
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Simplicial complexes on data sets:
- Vietoris Rips complex
- Cech complex
- The nerve construction
- Mapper
Interleaving.
Textbook: Chapter 5
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Algebraic groups. Intuition and idea of homology.
Textbook: Chapter 6, idea of chapter 7
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Homology groups
Textbook: Chapters 7 and 8.1
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Computing homology
Textbook: Chapters 7 and 8.1
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Persistent homology
Textbook: Chapter 9
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Filtrations, persistence modules, interleavings, Gromov-Hausdorff distance
Textbook: Chapters 9 and 10
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Stability theorem for persistent homology, Wasserstein distance, further developments of persistence, examples of applications
Textbook: Chapter 10
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Write to Santa...
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Discrete Morse theory: discrete Morse functions, discrete gradient vector fields
Textbook: Chapter 11 -