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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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

