Lec #  Date  Topic(s)  Scribe  Panopto 
1  Jan 12 
syllabus,
topology and connectivity, applications  drug
design, geography

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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

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3  Jan 19 
\(\mathbb{S}^1 \not\approx\) annulus, \(\mathbb{S}^2
\approx \mathbb{R}^2 \cup \{\infty\}\), 2manifold,
orientationreversing closed curve, nonorientable
manifold, 0 and 1manifold classifications

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4  Jan 21 
open cover, compact spaces, chart, Hausdorff,
\(d\)manifold, projective plane, Klein bottle,
classification of \(2\)manifolds, connected sum

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5  Jan 26 
combinations, linear (affine) independence,
\(k\)simplex, (co)face, simplcial complex \(K\),
underlying space, abstract simplicial complex (ASC)

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6  Jan 28 
general position, isomorphic ASCs, examples of ASCs
 cylinder, Möbius strip, torus, Klein bottle,
rule to check triangulations of 2manifolds

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7  Feb 2 
Euler characteristic \(\chi\),
\(\chi(\mathbb{T}^2)=\chi(\mathbb{K}^2)=0\), \(\chi
+\)orientation is complete invariant, genus, cross
cap, \(\chi(g\mathbb{T}^2)=22g\), orientation of
simplex

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8  Feb 4 
Homework 2
disucssion, induced orientation, consistently
oriented simplices, (non)orientable manifold,
propagating orientation

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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

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10  Feb 11 
simplicial complex as poset, \(\rm{St}(X)=X\) and
everything above, homotopy, deformation retract,
homotopy equivalence, contractible, nerve

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11  Feb 16 
Nerve theorem, Čech complex, \(r_i \leq r_j
\Rightarrow \rm{Čech}(r_i) \subseteq
\rm{Čech}(r_j)\), miniball, VietorisRips (VR)
complex, VR lemma: \(\rm{VR}(r) \subseteq
\rm{Čech}(\sqrt{2}r)\)

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12  Feb 18 
Voronoi diagram, Delaunay trinagulation (DT), DT in
Octave, general position, filtration, filtration
ordering, signature, alpha complex

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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

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14  Feb 25 
witness complex, \(W_{\infty}(L,S) \subseteq
\rm{Del}_L\), lazy witness complex, maxmin
selection, review of groups, identity, inverse,
subgroups

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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

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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, \(\partial_p \partial_{p+1} =
0\)

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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 \(\beta_p\)

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18  Mar 10 
betti number examples: torus, \(p\)ball,
EulerPoincaré theorem: \(\chi = \sum_p
(1)^p \beta_p\), boundary matrix, boundary as a
matrixvector product

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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 \(p\)

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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

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21  Mar 29 
persistence, lifetime of \(\sigma\),
positive/negative simplex, incremental algo for
\(\beta_j\) in \(\mathbf{S}^3\), example, checking
cycle membership (unionfind)

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22  Mar 31 
unionfind example, persistence algo, canonical
cycle, details of pairing
\((\sigma^i_+,\sigma^j_)\), youngest \(+\)simplex
in \(\Gamma\), indexpersistence diagram

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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

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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

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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)

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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

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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

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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 \(\lambda \rightarrow 0\)

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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

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30  Apr 28 
implementation of mapper, interval length and
overlap, clustering data points in each
\(f^{1}(U_{\alpha})\), simplicial complex
representation

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