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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| 15 |
Mar 3 |
\(p\)-chains and group \(C_p(K)\), elementary chains,
boundary \(\partial_p: C_p \to C_{p-1}\), examples
over \(\mathbb{Z}_2\) and \(\mathbb{Z}\), \(p\)-cycle
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Scb15
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Vid15
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| 16 |
Mar 5 |
\(p\)-boundary, 0-chain is a 0-cycle, fundamental
lemma of homology: \(\partial \partial \mathbf{d} =
0\), \(p\)-homology group: \(H_p = Z_p/B_p\)
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Scb16
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Vid16
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| 17 |
Mar 10 |
rank\(H_p\)\(=\)\(\beta_p\): \(p\)-th Betti number,
\(H_p\) of torus & \(p\)-ball, Euler-Poincaré
theorem \(\chi = \sum_p (-1)^p \beta_p\), boundary
matrix
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Scb17
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Vid17
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| 18 |
Mar 12 |
E(R/C)Os, Smith normal form, SNF\((\begin{bmatrix}
\partial_p \end{bmatrix}) = U_{p-1} \begin{bmatrix}
\partial_p \end{bmatrix} V_p \), SNF\((\begin{bmatrix}
\partial_p \end{bmatrix})\) gives \(z_p, b_{p-1}\),
bases for \(Z_p, B_{p-1}\), example
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Scb18
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Vid18
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| 19 |
Mar 24 |
SNF algorithm over \(\mathbb{Z}_2\), reduced homology,
augmentation map, \(\tilde{\beta}_0 = \beta_0-1\),
relative homology group \(H_p(K,K_0)\)
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Scb19
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Vid19
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| 20 |
Mar 26 |
persistent homology, creator/destroyer simplices,
persistence of feature, incremental algorithm for
\(\beta_k\)'s of \(K \subset \mathbb{S}^3\)
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Scb20
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Vid20
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| 21 |
Mar 31 |
incremental aglo
example, UNION-FIND
for cnctd cpnts, persistence algo, canonical cycle,
youngest +ve simplex
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Scb21
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Vid21
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| 22 |
Apr 2 |
persistence example, index-persistence diagram, count
triangles for \(\beta_k^{\ell,p}\), set \(T[i]=j\) for
pair \((\sigma^i, \sigma^j)\), collision
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Scb22
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Vid22
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| 23 |
Apr 7 |
persistence diagram, fundamental lemma of persistent
homology, radius/sublevel persistence, matrix
reduction algo
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Scb23
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Vid23
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| 24 |
Apr 9 |
video: lowest ones independent of reduction,
pairing lemma, persistence diagram, mapper, cover,
refined pullback
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Scb24
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Vid24
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| 25 |
Apr 14 |
video: mapper algo, filter/distance fns,
resolution, gain, Reeb graph, Morse theory, map of
coverings, examples
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Scb25
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Vid25
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| 26 |
Apr 16 |
video: homology over \(\mathbb{Z}\), optimal
homologous cycle problem (OHCP), \(H_1\) of
Möbius (\(M\)), \(\mathbb{R}P^2, \mathbb{K}^2\),
\(H_1(M,\) edge)
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Scb26
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Vid26
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| 27 |
Apr 21 |
video: torsion, \(|x_i|\)\(\to\)\(x_i^+ -
x_i^-\), OHCP as integer program, linear programming
(LP) intro, total unimodularity (TU)
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Scb27
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Vid27
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| 28 |
Apr 23 |
video: ops preserving TU, \(A\) of OHCP
TU\(\Leftrightarrow\)\(\begin{bmatrix} \partial
\end{bmatrix}\) TU, orientable
manifold\(\Rightarrow\)\(\begin{bmatrix} \partial
\end{bmatrix}\) TU, \([\partial_2]\) TU iff \(K\)
has no Möbius strip
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Scb28
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Vid28
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| 29 |
Apr 28 |
video: \([\partial(K)]\) TU iff \(K\) has no
relative torsion, \(H_1(k\)-fold dunce hat\() \simeq
\mathbb{Z}_k\), NTU Neutralized complex, optimal basis
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Scb29
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Vid29
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| 30 |
Apr 30 |
video: currents, flat norm (FN), FN distance
better than Hausdorff, Frechet, multiscale simplicial
flat norm (MSFN)
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Scb30
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Vid30
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Last modified: Mon Apr 27 22:05:28 PDT 2026
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