Lec #  Date  Topic(s)  Scribe  Panopto 
1  Jan 9 
syllabus,
topology and connectivity, examples of optimiztion
in topology: tunnels in drug design, terrain
smoothing in geography

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video

2  Jan 11 
no
class



3  Jan 16 
definition of topology, interior, closure, and
boundary of sets, homeomorphism, circle
\(\not\approx\) annulus, \(\mathbb{S}^2 \approx
\mathbb{R}^2 \cup \{\infty\}\)

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video

4  Jan 18 
2manifold (with boundary), (non)orientable
manifolds, 0, 1manifolds, compact, Hausdorff,
\(d\)manifold, torus, projective plane

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video

5  Jan 23 
connected sum, \(\mathbb{R}P^2 \# \mathbb{R}P^2
\approx \mathbb{K}^2\), simplex, (co)face,
simplicial complex, underlying space, abstract
simplicial complex (ASC)

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video

6  Jan 25 
geometric realization theorem, triangulation, ASCs
for surfaces, topological invariant, Euler
characteristic \(\chi\),
\(\chi(\mathbb{T}^2)=\chi(\mathbb{K}^2)=0\)

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video

7  Jan 30 
genus, cross cap, \(\chi(g\mathbb{T}^2)=22g,~
\chi(g\,\mathbb{R}P^2)=2g\), orientation of
simplex, comparing orientation, orientable manifold

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video

8  Feb 1 
checking orientability, propagating orientation,
subdivision, star St\(\,v\) and link Lk\(\,v\) of
vertex \(v\), St\(\,\sigma\) of \(\sigma \in K\), St
\(X\) of \(X \subset K\)

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video

9  Feb 6 
example of St \(X\), partial order, poset, principal
simplices, homotopy, retract, deformation retract,
homotopy equivalent, contractible

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video

10  Feb 8 
nerve theorem, Čech complex, \(r_i \leq r_j
\Rightarrow {\rm Čech}(r_i) \subseteq {\rm
Čech}(r_j)\), VietorisRips (VR) complex, VR
lemma: \({\rm VR}(r) \subseteq {\rm
Čech}(\sqrt{2}r)\)

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video

11  Feb 13 
Voronoi diagram, Delaunay complex, general position,
filtration, filtered complex, alpha complexes,
\(\rm{Alpha}(r) \subseteq \rm{Del},
\rm{Čech}(r)\),

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video

12  Feb 15 
power distance, power Voronoi/Delaunay complex,
weighted alpha complex, empty circumsphere property,
weak/strong witness

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video

13  Feb 20 
witness complex, \(W_{\infty}(L,S) \subseteq
\mbox{Del}_L\), laxy witness complex, random/maxmin
selection of landmarks, groups and homomorphisms

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video

14  Feb 22 
\(p\)chain, group of \(p\)chains \(C_p\),
elementary chain, boundary, \(p\)th boundary map
\(\partial_p: C_p \to C_{p1}\), chain complex,
\(p\)cycle

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video

15  Feb 27 
\(p\)boundary, half of 0cycles are 0boundaries,
\(\partial \partial \mathbf{d} = 0\), \(p\)homology
group \(H_p = Z_p/B_p\), ord\((C_p)\), rank\(H_p =
\beta_p\): \(p\)betti number

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video

16  Mar 1 
makeup
lecture: examples of homology groups: torus,
\(p\)ball, EulerPoincaré theorem: \(\chi =
\sum_p (1)^p \beta_p\), boundary matrix
\(\begin{bmatrix} \partial_p \end{bmatrix}\)

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video

17  Mar 6 
EROs and ECOs, Smith normal form,
SNF\((\begin{bmatrix} \partial_p \end{bmatrix}) =
U_{p1} \begin{bmatrix} \partial_p \end{bmatrix} V_p
\), SNF\((\begin{bmatrix} \partial_p
\end{bmatrix})\) gives \(z_p, b_{p1}\) and bases
for \(Z_p, B_{p1}\), example

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video

18  Mar 8 
bases for \(Z_1, B_1, H_1\) in example, SNF
algorithm over \(\mathbb{Z}_2\), reduced homology,
augmentation map, \(\tilde{\beta}_0 = \beta_01\),
relative homology

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video

19  Mar 20 
\(p\)persistent \(k\)homology group of
\(K^{\ell}\), persistence of \([\mathbb{z}]\),
creator and destroyer simplices, incremental
algorithm for \(\beta_k\)'s in \(\mathbb{S}^3\)

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video

20  Mar 22 
UNIONFIND
data structure, persistence algorithm, canonical
cycle, youngest positive simplex, example, details
of pairing

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video

21  Mar 27 
indexpersistence diagram, \(\beta_k^{\ell,p}\) by
counting triangles, implementation of pairing,
\(T[i]=j\) for pair \((\sigma^i, \sigma^j)\),
collision, illustration

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video

22  Mar 29 
lecture canceled



23  Apr 3 
more details of persistence, persistence diagram,
fundamental lemma of persistent homology, matrix
reduction, example

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video

24  Apr 5 
lowest ones independent of reduction, pairing lemma,
persistence diagram, mapper algorithm, pullback and
refined pullback

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video

25  Apr 10 
reeb graph, map of coverings, choices for covers in
mapper, homology over \(\mathbb{Z}\), optimal
homologous cycle problem (OHCP)

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video

26  Apr 12 
cover intervals in mapper, homology over
\(\mathbb{Z}\): Möbius strip, projective plane,
klein bottle, OHCP as an integer program (IP)

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video

27  Apr 17 
about Hw6, project, total
unimodularity (TU), OHCP LP and TU,
\(\begin{bmatrix} \partial_{p+1}(K) \end{bmatrix}\)
is TU for orientable \((p+1)\)manifold, MCM

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video

28  Apr 19 
\([\partial_2(K)]\) TU iff \(K\) has no Möbius
strip, \([\partial(K)]\) TU iff \(K\) has no
relative torsion, NTU Neutralized complex, currents,
flat norm

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video

29  Apr 24 
multiscale simplicial flat norm (MSFN), MSFN LP and
TU, integral decomposition of currents,
counterexample on Klein bottle

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video

30  Apr 26 
optimal bounding chain problem (OBCP), OBCP and
related OHCP, simplicial median shapes, integer
solutions for nonTU cases

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video

