Book Review
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MathSciNet review:
4038019
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Book Information:
Author:
Robert W. Ghrist
Title:
Elementary applied topology
Additional book information:
Createspace,
2014,
ISBN 978-1-5028-8085-7,
US$19.99
Author:
Steve Y. Oudot
Title:
Persistence theory: From quiver representations to data analysis
Additional book information:
Mathematical Surveys and Monographs, Vol. 209,
American Mathematical Society,
Providence, RI,
2015,
viii+218 pp.,
ISBN 978-1-4704-2545-6,
US$65.00
O. Bobrowski and M. Kahle, Topology of random geometric complexes: a survey, J. Appl. and Comput. Topology, (2018), 1–34.
Gunnar Carlsson and Vin de Silva, Zigzag persistence, Found. Comput. Math. 10 (2010), no. 4, 367–405. MR 2657946, DOI 10.1007/s10208-010-9066-0
Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian, On the local behavior of spaces of natural images, Int. J. Comput. Vis. 76 (2008), no. 1, 1–12. MR 3715451, DOI 10.1007/s11263-007-0056-x
Gunnar Carlsson and Facundo Mémoli, Characterization, stability and convergence of hierarchical clustering methods, J. Mach. Learn. Res. 11 (2010), 1425–1470. MR 2645457
Gunnar Carlsson and Afra Zomorodian, The theory of multidimensional persistence, Discrete Comput. Geom. 42 (2009), no. 1, 71–93. MR 2506738, DOI 10.1007/s00454-009-9176-0
Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan, Topology of viral evolution, Proc. Natl. Acad. Sci. USA 110 (2013), no. 46, 18566–18571. MR 3153945, DOI 10.1073/pnas.1313480110
F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot, Proximity of persistence modules and their diagrams, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, Association for Computing Machinery, New York, 2009, pp. 237–246.
Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot, The structure and stability of persistence modules, SpringerBriefs in Mathematics, Springer, [Cham], 2016. MR 3524869, DOI 10.1007/978-3-319-42545-0
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. MR 2279866, DOI 10.1007/s00454-006-1276-5
Carina Curto, What can topology tell us about the neural code?, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 1, 63–78. MR 3584098, DOI 10.1090/bull/1554
Carina Curto and Vladimir Itskov, Cell groups reveal structure of stimulus space, PLoS Comput. Biol. 4 (2008), no. 10, e1000205, 13. MR 2457124, DOI 10.1371/journal.pcbi.1000205
Y. Dabaghian, F. Mémoli, L, Frank, and G. Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Computational Biology 8 (2012), e1002581.
Harm Derksen and Jerzy Weyman, Quiver representations, Notices Amer. Math. Soc. 52 (2005), no. 2, 200–206. MR 2110070
Alireza Dirafzoon, Alper Bozkurt, and Edgar Lobaton, Geometric learning and topological inference with biobotic networks, IEEE Trans. Signal Inform. Process. Netw. 3 (2017), no. 1, 200–215. MR 3615702, DOI 10.1109/TSIPN.2016.2623093
Jean-Guillaume Dumas, Frank Heckenbach, David Saunders, and Volkmar Welker, Computing simplicial homology based on efficient Smith normal form algorithms, Algebra, geometry, and software systems, Springer, Berlin, 2003, pp. 177–206. MR 2011758
Emerson G. Escolar and Yasuaki Hiraoka, Persistence modules on commutative ladders of finite type, Discrete Comput. Geom. 55 (2016), no. 1, 100–157. MR 3439262, DOI 10.1007/s00454-015-9746-2
L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii academiae scientiarum Petropolitanae 8 (1741), 128–140.
Lisbeth Fajstrup, Martin Raußen, and Eric Goubault, Algebraic topology and concurrency, Theoret. Comput. Sci. 357 (2006), no. 1-3, 241–278. MR 2242768, DOI 10.1016/j.tcs.2006.03.022
Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221. MR 1957228, DOI 10.1007/s00454-002-0760-9
Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh, Confidence sets for persistence diagrams, Ann. Statist. 42 (2014), no. 6, 2301–2339. MR 3269981, DOI 10.1214/14-AOS1252
Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov, Clique topology reveals intrinsic geometric structure in neural correlations, Proc. Natl. Acad. Sci. USA 112 (2015), no. 44, 13455–13460. MR 3429279, DOI 10.1073/pnas.1506407112
C. Hierholzer, Ueber Kegelschnitte im Raume, Math. Ann. 2 (1870), no. 4, 563–586 (German). MR 1509680, DOI 10.1007/BF01444042
A. Maleki, M. Shahram, and G. Carlsson, A near optimal coder for image geometry with adaptive partitioning, in Proceedings of the 15th IEEE International Conference on Image Processing, Institute of Electrical and Electronics Engineers, Piscataway, NJ, 2008, pp. 1061–1064.
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod, and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science 6 (2017), no. 1, 17.
Jose A. Perea and Gunnar Carlsson, A Klein-bottle-based dictionary for texture representation, Int. J. Comput. Vis. 107 (2014), no. 1, 75–97. MR 3165575, DOI 10.1007/s11263-013-0676-2
Leonid Polterovich and Egor Shelukhin, Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules, Selecta Math. (N.S.) 22 (2016), no. 1, 227–296. MR 3437837, DOI 10.1007/s00029-015-0201-2
Jacob T. Schwartz and Micha Sharir, On the “piano movers’ ” problem. I. The case of a two-dimensional rigid polygonal body moving amidst polygonal barriers, Comm. Pure Appl. Math. 36 (1983), no. 3, 345–398. MR 697469, DOI 10.1002/cpa.3160360305
P. Škraba, Persistent homology and machine learning, Informatica 42 (2018), 2.
Steve Smale, On the topology of algorithms. I, J. Complexity 3 (1987), no. 2, 81–89. MR 907191, DOI 10.1016/0885-064X(87)90021-5
Michael Usher and Jun Zhang, Persistent homology and Floer-Novikov theory, Geom. Topol. 20 (2016), no. 6, 3333–3430. MR 3590354, DOI 10.2140/gt.2016.20.3333
Wei, G.-W. Persistent homology analysis of biomolecular data. Journal of Computational Physics 305, 276–299.
Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005), no. 2, 249–274. MR 2121296, DOI 10.1007/s00454-004-1146-y
References
- O. Bobrowski and M. Kahle, Topology of random geometric complexes: a survey, J. Appl. and Comput. Topology, (2018), 1–34.
- Gunnar Carlsson and Vin de Silva, Zigzag persistence, Found. Comput. Math. 10 (2010), no. 4, 367–405. MR 2657946, DOI 10.1007/s10208-010-9066-0
- Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian, On the local behavior of spaces of natural images, Int. J. Comput. Vis. 76 (2008), no. 1, 1–12. MR 3715451, DOI 10.1007/s11263-007-0056-x
- Gunnar Carlsson and Facundo Mémoli, Characterization, stability and convergence of hierarchical clustering methods, J. Mach. Learn. Res. 11 (2010), 1425–1470. MR 2645457
- Gunnar Carlsson and Afra Zomorodian, The theory of multidimensional persistence, Discrete Comput. Geom. 42 (2009), no. 1, 71–93. MR 2506738, DOI 10.1007/s00454-009-9176-0
- Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan, Topology of viral evolution, Proc. Natl. Acad. Sci. USA 110 (2013), no. 46, 18566–18571. MR 3153945, DOI 10.1073/pnas.1313480110
- F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot, Proximity of persistence modules and their diagrams, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, Association for Computing Machinery, New York, 2009, pp. 237–246.
- Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot, The structure and stability of persistence modules, SpringerBriefs in Mathematics, Springer, [Cham], 2016. MR 3524869, DOI 10.1007/978-3-319-42545-0
- David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. MR 2279866, DOI 10.1007/s00454-006-1276-5
- Carina Curto, What can topology tell us about the neural code?, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 1, 63–78. MR 3584098, DOI 10.1090/bull/1554
- Carina Curto and Vladimir Itskov, Cell groups reveal structure of stimulus space, PLoS Comput. Biol. 4 (2008), no. 10, e1000205, 13. MR 2457124, DOI 10.1371/journal.pcbi.1000205
- Y. Dabaghian, F. Mémoli, L, Frank, and G. Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Computational Biology 8 (2012), e1002581.
- Harm Derksen and Jerzy Weyman, Quiver representations, Notices Amer. Math. Soc. 52 (2005), no. 2, 200–206. MR 2110070
- Alireza Dirafzoon, Alper Bozkurt, and Edgar Lobaton, Geometric learning and topological inference with biobotic networks, IEEE Trans. Signal Inform. Process. Netw. 3 (2017), no. 1, 200–215. MR 3615702, DOI 10.1109/TSIPN.2016.2623093
- Jean-Guillaume Dumas, Frank Heckenbach, David Saunders, and Volkmar Welker, Computing simplicial homology based on efficient Smith normal form algorithms, Algebra, geometry, and software systems, Springer, Berlin, 2003, pp. 177–206. MR 2011758
- Emerson G. Escolar and Yasuaki Hiraoka, Persistence modules on commutative ladders of finite type, Discrete Comput. Geom. 55 (2016), no. 1, 100–157. MR 3439262, DOI 10.1007/s00454-015-9746-2
- L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii academiae scientiarum Petropolitanae 8 (1741), 128–140.
- Lisbeth Fajstrup, Martin Raußen, and Eric Goubault, Algebraic topology and concurrency, Theoret. Comput. Sci. 357 (2006), no. 1-3, 241–278. MR 2242768, DOI 10.1016/j.tcs.2006.03.022
- Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221. MR 1957228, DOI 10.1007/s00454-002-0760-9
- Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh, Confidence sets for persistence diagrams, Ann. Statist. 42 (2014), no. 6, 2301–2339. MR 3269981, DOI 10.1214/14-AOS1252
- Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov, Clique topology reveals intrinsic geometric structure in neural correlations, Proc. Natl. Acad. Sci. USA 112 (2015), no. 44, 13455–13460. MR 3429279, DOI 10.1073/pnas.1506407112
- C. Hierholzer, Ueber Kegelschnitte im Raume, Math. Ann. 2 (1870), no. 4, 563–586 (German). MR 1509680, DOI 10.1007/BF01444042
- A. Maleki, M. Shahram, and G. Carlsson, A near optimal coder for image geometry with adaptive partitioning, in Proceedings of the 15th IEEE International Conference on Image Processing, Institute of Electrical and Electronics Engineers, Piscataway, NJ, 2008, pp. 1061–1064.
- N. Otter, M. A. Porter, U. Tillmann, P. Grindrod, and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science 6 (2017), no. 1, 17.
- Jose A. Perea and Gunnar Carlsson, A Klein-bottle-based dictionary for texture representation, Int. J. Comput. Vis. 107 (2014), no. 1, 75–97. MR 3165575, DOI 10.1007/s11263-013-0676-2
- Leonid Polterovich and Egor Shelukhin, Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules, Selecta Math. (N.S.) 22 (2016), no. 1, 227–296. MR 3437837, DOI 10.1007/s00029-015-0201-2
- Jacob T. Schwartz and Micha Sharir, On the “piano movers’ ” problem. I. The case of a two-dimensional rigid polygonal body moving amidst polygonal barriers, Comm. Pure Appl. Math. 36 (1983), no. 3, 345–398. MR 697469, DOI 10.1002/cpa.3160360305
- P. Škraba, Persistent homology and machine learning, Informatica 42 (2018), 2.
- Steve Smale, On the topology of algorithms. I, J. Complexity 3 (1987), no. 2, 81–89. MR 907191, DOI 10.1016/0885-064X(87)90021-5
- Michael Usher and Jun Zhang, Persistent homology and Floer-Novikov theory, Geom. Topol. 20 (2016), no. 6, 3333–3430. MR 3590354, DOI 10.2140/gt.2016.20.3333
- Wei, G.-W. Persistent homology analysis of biomolecular data. Journal of Computational Physics 305, 276–299.
- Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005), no. 2, 249–274. MR 2121296, DOI 10.1007/s00454-004-1146-y
Review Information:
Reviewer:
José A. Perea
Affiliation:
Department of Computational Mathematics, Science & Engineering, Department of Mathematics, Michigan State University, East Lansing, Michigan
Email:
joperea@msu.edu
Journal:
Bull. Amer. Math. Soc.
57 (2020), 153-159
DOI:
https://doi.org/10.1090/bull/1647
Published electronically:
September 24, 2018
Additional Notes:
This work was partially supported by the NSF (DMS-1622301) and DARPA (HR0011-16-2-003)
Review copyright:
© Copyright 2018
American Mathematical Society
