Ralph Martin Kaufmann
Purdue University; Department of Mathematics
150 N. University Street; West Lafayette, IN
 Phone: (765) 494-1205; Fax: (765) 494-0548
e-mail: rkaufman@math.purdue.edu  URL: http://www.math.purdue.edu/~rkaufman

 
MA 59800  - Fall 2018
Topological Data Analysis



Instructor: Ralph Kaufmann

                    Office: Math 730
                     Contact via e-mail

Date and Time:   TR 9:00 - 10:15 pm in MATH 215
Syllabus

Office hours: Th 11:00- 11:50 and  by appointment 


References:  "Computational Topology" by Herbert Edelsbrunner and John L. Harer (CT)
                       "Topological pattern recognition for point cloud data" Acta Numerica. 23: 289-368 by Gunnar Carlsson. (TPR)
                       "Barcodes: The Persistent Topology of Data". Bull. Amer. Math. Soc. 45 (2008), 61-75 by Robert Ghrist.(Bar)
                        "Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition"
                           by G. Singh, F. Memoli and G. Carlsson (Mapper)
 

more tba.


First Week: I will do a short survey and assess the different needs and levels of the participants.

Date
 Topics
 Reference
8/21
 Overview. Topological space, continuous maps, homeomorphism, metric space, subspace, simplex
 Any topology book/wikipedia
8/23
 Groups, free groups, chains, homology
 Any topology book, algebra book, wikipedia
8/28
 Simplices, combinatorics, Singular Chains and Homology
Any topology book, e.g. Munkres,
Gelfand-Manin, wikipedia
8/30
 Categories and Functors, functoriality of chains and homology
Any topology book, e.g. Munkres,
Gelfand-Manin, wikipedia
9/4
 Simplicial category and simplicial objects, nerve of a category.
 Gelfand-Manin, other topology books wikipedia
9/6
 Simplicial chain as simplicial objects, realization functor, semi-simplicial sets, triangulated spaces
 Gelfand-Manin, other topology books, wikipedia
9/11
 Comparing Homology theories, computations of small examples.
 Munkres (esp. § 13 76pp, §34), lecture
9/14
 Comparing Homology theories, computations of small examples. Munkres (esp. § 13 76pp, §34), lecture
9/18
 Homology of graphs, Euler Characteristic, Chech complex, Vietoris-Rips complex Any topology book, wikipedia, CT + TPR
9/20
 Persistence spaces, filtered objects, Persistence Diagrams, Bar Codes
 CT+TPR + lecture
9/25
 Persistent homology, bar codes part II, examples
 CT+TPR+Bar + lecture
9/27
 Computation of normal form for matrices over Z Munkres (§ 11)
10/2
 Normal forms over R and extension of coefficients, Computation of homology
 lecture (good Lin alg book)+Munkres (§ 11)
10/4
 Algorithm for computing bar codes and Persistence diagrams and example TPR+ lecture
10/16
 Algorithm for computing bar codes and Persistence diagrams and example TPR+ lecture
10/18
 Algorithm for computing bar codes and Persistence diagrams and example
 TPR+ lecture
10/23
 Classification of finitely presented persistence vector spaces
 CT
10/25
 Remarks about last two lectures, Relative Homology, Long-Exact Sequence
 Lecture + any topology text.
10/30
 Axioms of Homology
 Mukres, or any other source
11/1
 Functorial persistence, Voronoi, Delaunay and Alpha complexes
 TPR, CT
11/6
 The space of barcodes and stability. Part I
 TPR, CT
11/8
 The space of barcodes and stability. Part II
 TPR, CT
11/13
 Switches, vines, vinyards; collapses and Hasse diagrams
 CT
11/15
 Nathanael (Cosmic Web), Lance (Application of TDA to texts)
 1)
11/27
 Negin and Sarah (Elevation for Protein Docking)
2)
11/29
 Andrew (Measures and Stability), Yiran (Gene Expression) 3)
12/4
 Victor (Application of TDA to music), Duy (Applications of TDA to neuroscience) 4)
12/6
 MAPPER
 Mapper
 

1) Local homology of word embeddings: https://arxiv.org/pdf/1810.10136.pdf

Github of the article: https://github.com/temcinas/applied_topology_project

The Vietoris-Rips complex: https://doi-org.ezproxy.lib.purdue.edu/10.1016/j.cag.2010.03.007

Word embedding paper: https://nlp.stanford.edu/pubs/glove.pdf
2) Topology of Viral Evolution,  http://www.pnas.org/content/110/46/18566
3) Robust Detection of Periodic Patterns in Gene Expression Microarray Data using Topological Signal Analysis https://arxiv.org/abs/1410.0608
4) Ren, I. Y. (2014). Complexity of musical patterns.  Retrieved from https://warwick.ac.uk/fac/cross_fac/complexity/study/emmcs/outcomes/studentprojects/ren_m1.pdf
Ren, I. Y., Chazal, F., & Del Genio, C. I. (2015). Topological data analysis on music networks.  Retrieved from https://warwick.ac.uk/fac/cross_fac/complexity/study/emmcs/outcomesstudentprojects/ren_m2.pdf
Sethares, W. A. & Budney, R. (2014). Topology of musical data. Journal of Mathematics and Music, 8, 73-92. doi:10.1080/17459737.2013.850597

Suggested Problems
Problem Set 1
Problem Set 2
Problem Set 3

Presentation Topics:
Ideally the presentation should be done in pairs of two. Otherwise, talk to me.
The length of the presentation should be at least about 45 min. Longer is no problem, basically until the hour is up.

NEWS: Presentations all went great. Thank you!