Computational Topology and Topological Data Analysis URI, Spring 2018


This is a reading seminar on aspects of applied algebraic topology and topological data analysis.

Meetings     Fridays noon, Tyler 052

Texts

1. Computational topology, Herbert Edelsbrunner and John L. Harer, AMS 2. Elements of Algebraic Topology, James R. Munkres, Addison-Wesley 3. Algebraic Topology, Allen Hatcher, Cambridge U. Press

Topics

1/26	0. Introduction		a. Introducing basic ideas of TDA b. Intro video: Introduction to persistent homology by Matthew Wright
2/2	1a. Graphs		a. Graphs b. Curves c. Knots, links
	1b. Basics of Topology		a. Topological spaces, metric space topology b. Maps: homeomorphisms, homotopy equivalence c. Manifolds notes: connected vs. path-connected spaces,   basic topology
2/9	2. Surfaces		a. 2-dimensional manifolds b. Triangulation notes: data structures (youtube)
2/16     2/23	3. Complexes		a. Simplicial complexes b. Convex set systems c. Dellaunay, Alpha complexes d. Vietoris-Rips complexes notes: A roadmap for the computation of persistent homology
3/2     3/9	4. Homology		a. Chains, boundaries, homology groups, betti numbers b. Induced maps among homology groups c. Homology groups d. Matrix reduction e. Relative homology f. Exact sequences
3/23     3/30	5. Duality		a. Cohomology b. Poincare, Alexander duality c. Intersection theory d. Alexander duality
4/6     4/13	7. Topological persistence		a. Filtrations, Persistent homology b. Persistence algorithm c. Persistence diagram, extended persistence d. Spectral sequences
4/6	9. Applications		a. Z. Cang, L. Mu, G. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
4/20	8. Stability		a. Matrix decomposition and updating b. Stability theorems c. Bipartite graph matching
4/27	9. Applications		a. Measures for gene expression data b. Elevation for protein docking c. Persistence for image segmentation
	6. Morse functions		a. Generic smooth functions b. Transversality c. Reeb graphs, approximating Reeb graphs from data
	10. Topology inference from data		a. Computing homology from data b. Sparsification to handle big data
	11. Computing optimized homology cycles		a. Computing shortest basis cycles on surfaces b. Computing shortest basis cycles from data points

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References

1. L. Wasserman, Topological data analysis
2. G. Carlsson, Topology and data
3. B. Brost, Computing Persistent Homology via Discrete Morse Theory
4. U. Fugacci, S. Scaramuccia, F. Iuricich, L. De Floriani, Persistent homology: a step-by-step introduction for newcomers
5. S. Weinberger, What is ... persistent homology?


6. Z. Cang, L. Mu, G. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening


7. N J Wildberger (UNSW), A course in Algebraic Topology


8. J.D. Boissonnat, C. S. Karthik, S. Tavenas, Building Efficient and Compact Data Structures for Simplicial Complexes
9. J. D. Boissonnat, C. Maria, The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

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