Text and class
material |
1. Computational topology, Herbert Edelsbrunner and John L. Harer, AMS
2. Curve and surface reconstruction: Algorithms with mathematical analysis, Tamal K. Dey, Cambridge U. Press
3. Elements of Algebraic Topology, James R. Munkres, Addison-Wesley
4. Algebraic Topology, Allen Hatcher, Cambridge U. Press
5. Class materials and notes posted on this web-site
|
Topics
1. Basics Topology
|
These
notes are much shortened versions of some chapters from an upcoming
book ``Computational Topology for Data Analysis" by myself and Y. Wang (UC SanDiego) to be published by Cambridge U. Press
Go here for the FRREE ELECTRONIC copy of the entire book.
|
a. Topological spaces, metric space topology [Notes]
|
b. Maps: homeomorphisms, homotopy equivalence, isotopy [Notes]
|
c. Manifolds [Notes]
[ClassScribble1] [ClassScribble2] [ClassScribble3]
|
2. Complexes on data
|
a. Simplicial complexes [Munkres][Notes]
b. Chech complexes, Vietoris-Rips complexes [Notes]
c. Witness complexes [deSilva-Carlsson04 paper][Notes]
d. Graph induced complexes [DeyFanWang13 paper][Notes]
[ClassScribble]
|
3. Homology
|
a. Chains, boundaries, homology groups, betti numbers [Notes, Munkres book]
c. Induced maps among homology groups [Notes, Munkres book]
d. Singular homology groups [Notes, Munkres book]
f. Cohomology groups [Notes, Hatcher book]
[ClassScribble]
|
4. Topological persistence
|
a. Filtrations, Persistent homology [Notes]
b. Persistence diagram [Notes] Cohen-SteinerEdelsbrunnerHarer07 paper proves the stabilty of persistence
c. Persistence algorithm [Notes] [C-VII Edelsbrunner-Harer book, EdelsbrunneLetscherZomorodian02 paper introduced topological persistence, ZomorodianCarlsson04 paper brings algebra into persistence]
[ClassScribble1] [ClassScribble2]
|
5. General Persistence (Zigzag)
|
a. Towers, Persistence modules from simplicial maps[Notes]
b. Algorithm for towers [Notes] [DeyFanWang13SM paper on Annotations]
c. Zigzag persistence and algorithms [Notes] [CarlssonSilvaMorozov09 paper on zigzag persistenc][DeyHou latest algo(2022)]
d. Level Set persistence [Notes]
e. Extended persistence [Notes, see the book]
[ClassScribble1][ClassScribble2]
|
6. Generators and Otimality
|
a. Computing optimal cycle basis [Notes] [EricksonWhittlesey05 paper on greedy basis construction and
b. Presentation slides, DeySunWang09 paper on shortest basis from point data]
c. Optimizing within a class [Notes][Presentation slides, DeyHiraniKrishnamoorthy10 paper on LP algorithm for shortest homologous cycle]
d. Computing optimal persistent cycles [Notes] [DeyHouMandal 2020 SODA paper]
|
7. Topology inference from point cloud data
|
a. Computing homology from data [Notes, ChazalOudot08 paper on homology inference, CCGGO09 paper on interleaving of persistence modules]
b. Sparsification to handle big data [Notes] [Presentation slides], [Sheehy12 paper on sparsified Rips complex, DeyFanWang13 paper on subsampling]
c. Homology inference [Notes]
d. Persistence diagram approximation from scalar data[Notes]
|
8. Persistence on graphs and Reeb graphs
|
a. Reeb graphs [Notes]
b. Interleaving distance [Notes]
c. Comparing graphs with persistence summary [Notes]
|
9. Discrete Morse Theory and Persistence
|
a. Discrete Morse [Notes]
b. Discrete Morse Vector Field (DMVF) [Notes]
c. Persistence Based DMVF [Notes]
d. Application to graph reconstruction [Notes]
|
10. Nerves, Mapper, Multiscale Mapper
|
a. Nerves [Notes]
b. Mapper [Notes]
c. Multiscale mapper [Notes]
|
11. Multiparameter persistence module decomposition
|
a. Multiparameter persistence [Notes]
b. Computing indecomposables [Notes]
c. Invariants [Notes]
|
12. Multiparameter persistence and distances
|
a. Multiparameter persistence module from categorial viewpoint [Notes]
b. Computing matching distance [Notes]
c. Computing interleaving distance [Notes]
|