Computational Topology - Lecture Notes and Videos
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Math 574 (Spring 2016)  - Lecture Notes and Videos on Computational Topology

Scribes from all lectures so far (as a single big file)

Lec # Date Topic(s) Scribe Panopto
1 Jan 12 syllabus, topology and connectivity, applications - drug design, geography scribe video
2 Jan 14 topological space $$\mathbb{X} = (X,T)$$, interior $$\mathring{A}$$, closure $$\bar{A}$$, and boundary $$\partial A$$ of $$A \in \mathbb{X}$$, continuous funtion, homeomorphism, examples scribe video
3 Jan 19 $$\mathbb{S}^1 \not\approx$$ annulus, $$\mathbb{S}^2 \approx \mathbb{R}^2 \cup \{\infty\}$$, 2-manifold, orientation-reversing closed curve, nonorientable manifold, 0- and 1-manifold classifications scribe video
4 Jan 21 open cover, compact spaces, chart, Hausdorff, $$d$$-manifold, projective plane, Klein bottle, classification of $$2$$-manifolds, connected sum scribe video
5 Jan 26 combinations, linear (affine) independence, $$k$$-simplex, (co)face, simplcial complex $$K$$, underlying space, abstract simplicial complex (ASC) scribe video
6 Jan 28 general position, isomorphic ASCs, examples of ASCs - cylinder, Möbius strip, torus, Klein bottle, rule to check triangulations of 2-manifolds scribe video
7 Feb  2 Euler characteristic $$\chi$$, $$\chi(\mathbb{T}^2)=\chi(\mathbb{K}^2)=0$$, $$\chi +$$orientation is complete invariant, genus, cross cap, $$\chi(g\mathbb{T}^2)=2-2g$$, orientation of simplex scribe video
8 Feb  4 Homework 2 disucssion, induced orientation, consistently oriented simplices, (non-)orientable manifold, propagating orientation scribe video
9 Feb  9 subdivision, barycentric subdivision, subcomplex, star, closed star, link of (St, $$\rm{Cl}\,\rm{St}$$, Lk) $$v \in K^0, \sigma \in K$$, and $$X \subseteq K$$, partial order, poset scribe video
10 Feb 11 simplicial complex as poset, $$\rm{St}(X)=X$$ and everything above, homotopy, deformation retract, homotopy equivalence, contractible, nerve scribe video
11 Feb 16 Nerve theorem, Čech complex, $$r_i \leq r_j \Rightarrow \rm{Čech}(r_i) \subseteq \rm{Čech}(r_j)$$, miniball, Vietoris-Rips (VR) complex, VR lemma: $$\rm{VR}(r) \subseteq \rm{Čech}(\sqrt{2}r)$$ scribe video
12 Feb 18 Voronoi diagram, Delaunay trinagulation (DT), DT in Octave, general position, filtration, filtration ordering, signature, alpha complex scribe video
13 Feb 23 $$\rm{Alpha}(r) \subseteq \rm{Del}, \rm{Čech}(r)$$, weighted/power distance, weighted alpha complex, empty circumsphere property of $$\rm{Del}$$, weak and strong witness scribe video
14 Feb 25 witness complex, $$W_{\infty}(L,S) \subseteq \rm{Del}_L$$, lazy witness complex, maxmin selection, review of groups, identity, inverse, subgroups scribe video
15 Mar  1 cosets, homomorphism, (epi/mono/iso)morphism, chains, group of $$p$$-chains $$C_p(K,\mathbf{Z}_2)$$ or $$C_p(K,\mathbf{Z})$$, elementary chains, $$p$$-chain as a vector scribe video
16 Mar  3 boundary of a $$p$$-simplex, $$\partial \mathbf{c} = \sum_i a_i \partial \sigma_i$$, cycle and boundary groups $$Z_p(K),B_p(K)$$, half of all $$0$$-cycles are $$0$$-boundaries, $$\partial_p \partial_{p+1} = 0$$ scribe video
17 Mar  8 relations of $$C_p, Z_p, B_p$$ over all $$p$$, homology group $$H_p = Z_p / B_p$$, examples, order$$(C_p)$$ and rank$$(C_p)$$ over $$\mathbf{Z}_2$$, betti numbers $$\beta_p$$ scribe video
18 Mar 10 betti number examples: torus, $$p$$-ball, Euler-Poincaré theorem: $$\chi = \sum_p (-1)^p \beta_p$$, boundary matrix, boundary as a matrix-vector product scribe video
19 Mar 22 matrix representations of elementary operations, Smith normal form (SNF), computing $$z_p,b_p,\beta_p$$ by reduction to SNF of $$[\partial_p]$$ for all $$p$$ scribe video
20 Mar 24 reduction algorithm for SNF, reduced homology groups, $$\tilde{\beta}_0 = \beta_0 - 1$$, relative homology, tracking $$\beta_1^{\ell}$$ over a filtration $$\{K^{\ell}\}_0^m$$ of gramicidin scribe video
21 Mar 29 persistence, lifetime of $$\sigma$$, positive/negative simplex, incremental algo for $$\beta_j$$ in $$\mathbf{S}^3$$, example, checking cycle membership (union-find) scribe video
22 Mar 31 union-find example, persistence algo, canonical cycle, details of pairing $$(\sigma^i_+,\sigma^j_-)$$, youngest $$+$$-simplex in $$\Gamma$$, index-persistence diagram scribe video
23 Apr   5 $$\beta_k^{\ell,p}=$$ # $$k$$-triangles containing $$(\ell,p)$$, details of pairing, set $$T[i]:=j$$ for pair $$(\sigma^i,\sigma^j)$$, finding youngest positive $$\sigma$$ in $$\Gamma$$, collision, illustration scribe video
24 Apr   7 homology over $$\mathbf{Z}$$, optimal homologous chain problem (OHCP), $$H_1(K,\mathbf{Z})$$ for $$K=$$Möbius, $$\mathbf{R}P^2, \mathbf{K}^2$$, torsion subgroup, torsion coefficients scribe video
25 Apr 12 $$\mathbf{K}^2$$ over $$\mathbf{Z}:\beta_1 = 1, t_1 =2$$, OHCP as an integer program (IP), optimization intro, rounding may not work for IP, total unimodularity (TU) scribe video
26 Apr 14 $$A$$ of OHCP LP is TU iff $$B=[\partial]$$ is, $$B$$ is TU when $$K$$ is orientable manifold, Möbius cycle matrix, visualizing torsion, $$k$$-fold dunce hat scribe video
27 Apr 19 option for project (data), cases of TU $$[\partial]$$, orientable manifolds, $$[\partial_d]$$ for $$K \in \mathbb{R}^d$$, NTU neutralized complexes, currents, flat norm scribe video
28 Apr 21 simplicial flat norm (SFN), as an LP, minimal surface problem on $$\mathbb{K}^2$$, optimal bounding chain problem (OBCP) $$\equiv$$ OHCP, SFN as $$\lambda \rightarrow 0$$ scribe video
29 Apr 26 Reeb graph $$R(f)$$ for $$(\mathbb{X},f)$$, Reeb graph for Morse functions, cover of $$\mathbb{X}$$ using $$f^{-1}(U_{\alpha})$$ for cover $$\{U_{\alpha}\}$$ of $$Z$$, for $$f:\mathbb{X} \rightarrow Z$$, examples scribe video
30 Apr 28 implementation of mapper, interval length and overlap, clustering data points in each $$f^{-1}(U_{\alpha})$$, simplicial complex representation scribe video

Last modified: Sat Jul 29 11:35:18 PDT 2017