Computational Topology: Lecture Notes and Videos

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 Panopto
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