Computational Topology - Lecture Notes and Videos

Math 574 (Spring 2018)  - 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  9 syllabus, topology and connectivity, examples of optimiztion in topology: tunnels in drug design, terrain smoothing in geography scribe video
2 Jan 11 no class
3 Jan 16 definition of topology, interior, closure, and boundary of sets, homeomorphism, circle $$\not\approx$$ annulus, $$\mathbb{S}^2 \approx \mathbb{R}^2 \cup \{\infty\}$$ scribe video
4 Jan 18 2-manifold (with boundary), (non-)orientable manifolds, 0-, 1-manifolds, compact, Hausdorff, $$d$$-manifold, torus, projective plane scribe 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) scribe video
6 Jan 25 geometric realization theorem, triangulation, ASCs for surfaces, topological invariant, Euler characteristic $$\chi$$, $$\chi(\mathbb{T}^2)=\chi(\mathbb{K}^2)=0$$ scribe video
7 Jan 30 genus, cross cap, $$\chi(g\mathbb{T}^2)=2-2g,~ \chi(g\,\mathbb{R}P^2)=2-g$$, orientation of simplex, comparing orientation, orientable manifold scribe 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 $$X \subset K$$ scribe video
9 Feb   6 example of St $$X$$, partial order, poset, principal simplices, homotopy, retract, deformation retract, homotopy equivalent, contractible scribe video
10 Feb   8 nerve theorem, Čech complex, $$r_i \leq r_j \Rightarrow {\rm Čech}(r_i) \subseteq {\rm Čech}(r_j)$$, Vietoris-Rips (VR) complex, VR lemma: $${\rm VR}(r) \subseteq {\rm Čech}(\sqrt{2}r)$$ scribe video
11 Feb 13 Voronoi diagram, Delaunay complex, general position, filtration, filtered complex, alpha complexes, $$\rm{Alpha}(r) \subseteq \rm{Del}, \rm{Čech}(r)$$, scribe video
12 Feb 15 power distance, power Voronoi/Delaunay complex, weighted alpha complex, empty circumsphere property, weak/strong witness scribe video