Math 529: Lecture Notes and Videos on Computational Topology
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Copyright: I (Bala Krishnamoorthy) hold the copyright
for all lecture scribes/notes, documents, and other
materials including videos posted on these course web
pages. These materials might not be used for commercial
purposes without my consent.
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Scribes from all lectures
so far (as single big file)
| Lec | Date | Topic(s) | Scribe | Video |
| 1 |
Jan 13 |
syllabus,
connected spaces,
applications: patient
trajectories, interface
features in chemistry, discrete connectivity
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Scb1
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Vid1
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| 2 |
Jan 15 |
topology, open sets, interior, closure, and boundary;
functions, homeomorphism, open disc \(\approx
\mathbb{R}^2\), circle \(\not\approx\) annulus
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Scb2
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Vid2
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| 3 |
Jan 20 |
\(\mathbb{S}^2 \approx \mathbb{R}^2 \cup \{\infty\}\),
stereographic projection, 2-manifold (with boundary),
non/orientable manifolds, 0-, 1-manifolds
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Scb3
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Vid3
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| 4 |
Jan 22 |
finite subcover, compact, Hausdorff, completely separable,
\(d\)-manifold, embedding, \(\mathbb{T}^2, \mathbb{R}P^2,
\mathbb{K}^2\), connected sum
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Scb4
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Vid4
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| 5 |
Jan 27 |
classification of 2-manifolds, \(\mathbb{R}P^2 \, \#
\, \mathbb{R}P^2 \approx \mathbb{K}^2\), simplex,
face, simplicial complex, dimension, underlying space
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Scb5
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Vid5
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| 6 |
Jan 29 |
abstract simplicial complex (ASC), geometric
realization of \(d\)-complex in \(\mathbb{R}^{2d+1}\),
triangulation, ASCs of surfaces
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Scb6
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Vid6
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| 7 |
Feb 3 |
topological invariant, Euler characteristic,
\(\chi(\mathbb{M}_1 \#
\mathbb{M}_2)\)\(=\)\(\chi(\mathbb{M}_1)+\chi(\mathbb{M}_2)-2\),
\(\chi+\)orientability: complete invariant
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Scb7
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Vid7
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| 8 |
Feb 5 |
genus, cross cap, using
\(\chi(g\mathbb{R}P^2)\)\(=\)\(2-g\), orientation of a
simplex, comparing orientations, orientable manifold
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Scb8
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Vid8
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| 9 |
Feb 10 |
propagating orientation, (barycentric) subdivision,
star St\(\,v\), closed star Cl St\(\,v\), link
Lk\(\,v\) of vertex \(v\), St\(\,\sigma\) of
\(\sigma\)\(\in\)\(K\)
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Scb9
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Vid9
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| 10 |
Feb 12 |
St \(X\) of \(X \subset K\), Lk\(\,X\) example, poset
representation of \(K\), principal simplices,
homotopy, deformation retraction
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Scb10
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Vid10
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| 11 |
Feb 17 |
nerve (Nrv), Nrv is ASC, nerve theorem, Čech and
Vietoris-Rips (VR) complexes, VR Lemma: \({\rm
VR}(r)\)\(\subseteq\)\({\rm Čech}(\sqrt{2}r)\)
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Scb11
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Vid11
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| 12 |
Feb 19 |
proof of VR lemma, Voronoi diagram, Delaunay complex,
general position, delaunay+voronoi in Matlab,
filtration
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Scb12
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Vid12
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| 13 |
Feb 24 |
alpha complex, \(\rm{Alpha}(r) \subseteq \rm{Del},
\rm{Čech}(r)\), weighted alpha complex, empty
sphere property, weak/strong witness
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Scb13
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Vid13
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| 14 |
Feb 26 |
strict witness complex, \(W_{\infty}(L,S) \subseteq
\mbox{Del}_L\), lazy witness complex, groups,
subgroups, cosets, homomorphisms
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Scb14
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Vid14
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Last modified: Thu Feb 26 16:10:18 PST 2026
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