Advertisement

Topological data analysis and cosheaves

  • Justin Michael CurryEmail author
Original Paper Area 1

Abstract

This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level set persistence is presented using sheaves and cosheaves.

Keywords

Topological data analysis Persistent homology Sheaves and cosheaves Barcodes o-minimal topology 

Mathematics Subject Classification

55U99 46M20 32S60 16G20 62-07 03C64 

References

  1. 1.
    Artin, M.: Algebra. Prentice Hall, Englewood Cliffs (1991). http://books.google.com/books?id=C_juAAAAMAAJ
  2. 2.
    Bendich, P., Marron, J., Miller, E., Pieloch, A., Skwerer, S.: Persistent homology analysis of brain artery trees (2014, preprint). arXiv:1411.6652
  3. 3.
    Boczko, E.M., Cooper, T.G., Gedeon, T., Mischaikow, K., Murdock, D.G., Pratap, S., Wells, K.S.: Structure theorems and the dynamics of nitrogen catabolite repression in yeast. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 16, pp. 5647–5652 (2005). doi: 10.1073/pnas.0501339102. http://www.pnas.org/content/102/16/5647.abstract
  4. 4.
    Bott, R.: Morse theory indomitable. Publications Mathématiques de l’IHÉS 68(1), 99–114 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bredon, G.: Sheaf theory. In: Axler S, Ribet K (eds) Graduate Texts in Mathematics, vol. 170, 2nd edn. Springer, Berlin (1997)Google Scholar
  6. 6.
    Bredon, G.E.: Topology and Geometry, vol. 139. Springer Science & Business Media, Berlin (1993)Google Scholar
  7. 7.
    Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 1–31 (2014). doi: 10.1007/s10208-014-9229-5
  8. 8.
    Carlsson, G., Ishkhanov, T., De Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)CrossRefGoogle Scholar
  9. 9.
    Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010). arXiv:0812.0197
  10. 10.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the Annual Symposium on Computational Geometry, pp. 247–256 (2009). http://www.mrzv.org/publications/zigzags/
  11. 11.
    Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discret. Comput. Geom. 42(1), 71–93 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.: Persistence barcodes for shapes. In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 124–135. ACM, New York (2004)Google Scholar
  13. 13.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discret. Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Coste, M.: An Introduction to Semialgebraic Geometry. Universite de Rennes (2002)Google Scholar
  15. 15.
    Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules (2012, preprint). arXiv:1210.0819
  16. 16.
    Curry, J.: Sheaves, cosheaves and applications. Ph.D. thesis, University of Pennsylvania (2014) [Publication number on Proquest is 3623819]Google Scholar
  17. 17.
    Curry, J., Ghrist, R., Nanda, V.: Discrete Morse theory for computing cellular sheaf cohomology. arXiv e-prints (2013)Google Scholar
  18. 18.
    Derkden, H., Weyman, J.: Quiver representations. Not. AMS 52(2), 200–206 (2005)Google Scholar
  19. 19.
    van den Dries, L.: Tame topology and O-minimal structures. In: London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1998). http://books.google.com/books?id=CLnElinpjOgC
  20. 20.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000 pp. 454–463. IEEE Redondo Beach, CA (2000)Google Scholar
  21. 21.
    Eilenberg, S., MacLane, S.: General theory of natural equivalences. Trans. Am. Math. Soc. 58(2), 231–294 (1945). http://www.jstor.org/stable/1990284
  22. 22.
    Goresky, R.M.: Triangulation of stratified objects. Proc. Am. Math. Soc., pp. 193–200 (1978)Google Scholar
  23. 23.
    Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002). http://books.google.com/books?id=BjKs86kosqgC
  24. 24.
    Iversen, B.: Cohomology of sheaves. Universitext. Springer, Berlin (1986). http://books.google.com/books?id=0R-9ngEACAAJ
  25. 25.
    Loi, T.: Verdier and strict thom stratifications in o-minimal structures. Ill. J. Math. 42(2), 347–356 (1998)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lum P.Y., Singh G., Lehman A., Ishkanov T., Vejdemo-Johansson M., Alagappan M., Carlsson J., Carlsson G.: Extracting insights from the shape of complex data using topology. Sci. Rep. 3 (2013). doi: 10.1038/srep01236
  27. 27.
    MacPherson, R., Schweinhart, B.: Measuring shape with topology. J. Math. Phys. 53(7), 073516 (2012). doi: 10.1063/1.4737391
  28. 28.
    Mather, J.: Notes on topological stability. Bull. (New Ser.) Am. Math. Soc. 49(4), 475–506 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    McCleary, J.: A User’s Guide to Spectral Sequences. In: Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001). http://books.google.com/books?id=NijkPwesh-EC
  30. 30.
    Munkres, J.R.: Elements of Algebraic Topology. Advanced book classics. Perseus Books (1984)Google Scholar
  31. 31.
    Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. 108(17), 7265–7270 (2011)CrossRefGoogle Scholar
  32. 32.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7(339–358), 24 (2007)Google Scholar
  33. 33.
    Spanier, E.: Algebraic Topology. Springer, Berlin (1994) [Originally published by McGraw-Hill in 1966]Google Scholar
  34. 34.
    Strang, G.: The fundamental theorem of linear algebra. Am. Math. Mon. 100(9), 848–855 (1993)Google Scholar
  35. 35.
    Treumann, D.: Exit paths and constructible stacks. Compositio Math. 145, 1504–1532 (2009). arXiv:0708.0659v1
  36. 36.
    Weibel, C.A.: An introduction to homological algebra. In: Garling D.J.H., Tom Dieck T., Walters I. (eds) Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)Google Scholar
  37. 37.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations