Math 529 (Spring 2024): Lecture Notes and Videos on Computational Topology
|
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.
|
Scribes from
all lectures so far (as a single big file)
| Lec | Date | Topic(s) | Scribe | Video |
| 1 |
Jan 9 |
syllabus,
connected spaces, applications: patient
trajectories, interface
features in chemistry, discrete connectivity
|
Scb1
|
Vid1
|
| 2 |
Jan 11 |
topology, open sets, interior, closure, and boundary;
functions, homeomorphism, open disc \(\approx
\mathbb{R}^2\), circle \(\not\approx\) annulus
|
Scb2
|
Vid2
|
| 3 |
Jan 16 |
\(\mathbb{S}^2 \approx \mathbb{R}^2 \cup \{\infty\}\),
stereographic projection, 2-manifold (with boundary),
non/orientable manifolds, 0-, 1-manifolds
|
Scb3
|
Vid3
|
| 4 |
Jan 18 |
finite subcover, compact, Hausdorff, completely separable,
\(d\)-manifold, embedding, \(\mathbb{T}^2, \mathbb{R}P^2,
\mathbb{K}^2\), connected sum
|
Scb4
|
Vid4
|
| 5 |
Jan 23 |
classification of 2-manifolds, simplex, face, simplicial
complex, underlying space, abstract simplicial complex (ASC)
|
Scb5
|
Vid5
|
| 6 |
Jan 25 |
geometric realization theorem: can realize \(d\)-complex in
\(\mathbb{R}^{2d+1}\), triangulation, ASCs of surfaces,
topological invariant
|
Scb6
|
Vid6
|
| 7 |
Jan 30 |
Euler characteristic \(\chi\), \(\chi(\mathbb{M}_1 \#
\mathbb{M}_2)\)\(=\)\(\chi(\mathbb{M}_1)+\chi(\mathbb{M}_2)-2\),
\(\chi+\)orientability: complete invariant, genus, cross
cap
|
Scb7
|
Vid7
|
| 8 |
Feb 1 |
using \(\chi(g\mathbb{R}P^2)\)\(=\)\(2-g\), orientation of simplex,
comparing orientations, orientable manifold, propagating
orientation
|
Scb8
|
Vid8
|
| 9 |
Feb 6 |
subdivision, barycentric subdivision, star St\(\,v\), closed
star Cl St\(\,v\), link Lk\(\,v\) of vertex \(v\), St\(\,\sigma\)
of \(\sigma \in K\), St\(X\) of \(X\)\(\subset\)\(K\)
|
Scb9
|
Vid9
|
| 10 |
Feb 8 |
Lk\(\,X\) example, partial order, poset representation,
principal simplices, homotopy, deformation retraction, nerve
|
Scb10
|
Vid10
|
| 11 |
Feb 13 |
nerve theorem, Čech complex, Vietoris-Rips (VR)
complex, VR Lemma: \({\rm VR}(r) \subseteq {\rm
Čech}(\sqrt{2}r)\), Delaunay complex
|
Scb11
|
Vid11
|
| 12 |
Feb 15 |
delaunay(n), voronoi fns in Matlab, filtration, alpha
complex, \(\rm{Alpha}(r) \subseteq \rm{Del},
\rm{Čech}(r)\), weighted alpha complex
|
Scb12
|
Vid12
|
| 13 |
Feb 20 |
empty sphere property, weak/strong witness, witness complex,
\(W_{\infty}(L,S) \subseteq \mbox{Del}_L\), lazy witness
complex, groups
|
Scb13
|
Vid13
|
| 14 |
Feb 22 |
subgroups, cosets, homomorphisms, \(p\)-chains and group
\(C_p(K)\), elementary chains, boundary \(\partial_p: C_p
\to C_{p-1}\)
|
Scb14
|
Vid14
|
| 15 |
Feb 27 |
\(p\)-cycle, \(p\)-boundary, 0-chain is 0-cycle, examples,
fundamental lemma: \(\partial \partial \mathbf{d} = 0\),
\(p\)-homology group \(H_p = Z_p/B_p\)
|
Scb15
|
Vid15
|
| 16 |
Feb 29 |
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
|
Scb16
|
Vid16
|
| 17 |
Mar 5 |
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
|
Scb17
|
Vid17
|
| 18 |
Mar 7 |
SNF algorithm over \(\mathbb{Z}_2\), reduced homology,
augmentation map, \(\tilde{\beta}_0 = \beta_0-1\), relative
homology group \(H_p(K,K_0)\)
|
Scb18
|
Vid18
|
| 19 |
Mar 19 |
persistent homology, creator/destroyer simplices,
persistence of a feature, incremental algorithm for
\(\beta_k\)'s of \(K \subset \mathbb{S}^3\)
|
Scb19
|
Vid19
|
| 20 |
Mar 21 |
On
Zoom: incremental aglo example, UNION-FIND,
persistence algo, canonical cycle, youngest positive simplex
|
Scb20
|
Vid20
|
| 21 |
Mar 26 |
persistence example, index-persistence diagram, count
triangles for \(\beta_k^{\ell,p}\), pairing: \(T[i]=j\) for
pair \((\sigma^i, \sigma^j)\), collision
|
Scb21
|
Vid21
|
| 22 |
Mar 28 |
persistence diagram, fundamental lemma of persistent
homology, radius/sublevel persistence, matrix reduction algo
|
Scb22
|
Vid22
|
| 23 |
Apr 2 |
lowest ones independent of reduction, pairing lemma, persistence
diagram, mapper, cover, nerve of refined pullback
|
Scb23
|
Vid23
|
| 24 |
Apr 4 |
mapper algorithm, filter/distance functions, resolution,
gain, Reeb graph, Morse theory, map of coverings, examples
|
Scb24
|
Vid24
|
| 25 |
Apr 9 |
homology over \(\mathbb{Z}\), optimal homologous cycle/chain
problem (OHCP), \(H_1\) of Möbius (\(M\)),
\(\mathbb{R}P^2, \mathbb{K}^2\), \(H_1(M,\) edge)
|
Scb25
|
Vid25
|
| 26 |
Apr 11 |
torsion, \(|x_i|\)\(\to\)\(x_i^+ - x_i^-\), OHCP as integer
program, linear programming (LP) for dummies, total
unimodularity (TU)
|
Scb26
|
Vid26
|
| 27 |
Apr 16 |
ops preserving TU, \(A\) of OHCP
TU\(\Leftrightarrow\)\(\begin{bmatrix} \partial
\end{bmatrix}\) TU, \(\begin{bmatrix} \partial
\end{bmatrix}\) TU for orientable manifold,
\([\partial_2(K)]\) TU iff \(K\) has no Möbius strip
|
Scb27
|
Vid27
|
| 28 |
Apr 18 |
\([\partial(K)]\) TU iff \(K\) has no relative torsion,
\(H_1(k\)-fold dunce hat\() \simeq \mathbb{Z}_k\), NTU Neutralized
complex, optimal basis
|
Scb28
|
Vid28
|
| 29 |
Apr 23 |
currents, flat norm (FN), FN distance better than Hausdorff,
Frechet, multiscale simplicial flat norm (MSFN) LP & TU
|
Scb29
|
Vid29
|
| 30 |
Apr 25 |
simplicial deformation theorem, integral decomposition of
currents & TU, median shapes LP not TU even when
\([\partial]\) is
|
Scb30
|
Vid30
|
Last modified: Thu Apr 25 17:48:18 PDT 2024
|