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Topological Persistence in Geometry and Analysis
About this Title
Leonid Polterovich, Tel Aviv University, Tel Aviv, Israel, Daniel Rosen, Ruhr-Universität Bochum, Bochum, Germany, Karina Samvelyan, Tel Aviv University, Tel Aviv, Israel and Jun Zhang, Université de Montréal, Montréal, Canada
Publication: University Lecture Series
Publication Year:
2020; Volume 74
ISBNs: 978-1-4704-5495-1 (print); 978-1-4704-5679-5 (online)
DOI: https://doi.org/10.1090/ulect/074
Table of Contents
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Front/Back Matter
A primer of persistence modules
- Definition and first examples
- Barcodes
- Proof of the isometry theorem
- What can we read from a barcode?
Applications to metric geometry and function theory
Persistent homology in symplectic geometry
- A concise introduction to symplectic geometry
- Hamiltonian persistence modules
- Symplectic persistence modules
- Peter Albers and Urs Frauenfelder, Square roots of Hamiltonian diffeomorphisms, J. Symplectic Geom. 12 (2014), no. 3, 427–434. MR 3248664
- D. Alvarez-Gavela, V. Kaminker, A. Kislev, K. Kliakhandler, A. Pavlichenko, L. Rigolli, D. Rosen, O. Shabtai, B. Stevenson, and J. Zhang, Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms, J. Topol. Anal. 11 (2019), no. 2, 467–498. MR 3958929, DOI 10.1142/S1793525319500213
- V. I. Arnol′d, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967), 1–14 (Russian). MR 0211415
- Gorô Azumaya, Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124. MR 37832
- Augustin Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), no. 2, 174–227 (French). MR 490874, DOI 10.1007/BF02566074
- S. A. Barannikov, The framed Morse complex and its invariants, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 93–115. MR 1310596
- Ulrich Bauer, Carsten Lange, and Max Wardetzky, Optimal topological simplification of discrete functions on surfaces, Discrete Comput. Geom. 47 (2012), no. 2, 347–377. MR 2872542, DOI 10.1007/s00454-011-9350-z
- Ulrich Bauer and Michael Lesnick, Induced matchings and the algebraic stability of persistence barcodes, J. Comput. Geom. 6 (2015), no. 2, 162–191. MR 3333456, DOI 10.20382/jocg.v6i2a9
- Paul Biran, Leonid Polterovich, and Dietmar Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003), no. 1, 65–118. MR 1991647, DOI 10.1215/S0012-7094-03-11913-4
- Sergey Bobkov and Michel Ledoux, One-dimensional empirical measures, order statistics, and Kantorovich transport distances, Mem. Amer. Math. Soc. 261 (2019), no. 1259, v+126. MR 4028181, DOI 10.1090/memo/1259
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Peter Bubenik, Vin de Silva, and Vidit Nanda, Higher interpolation and extension for persistence modules, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 272–284. MR 3683688, DOI 10.1137/16M1100472
- Lev Buhovsky, Vincent Humilière, and Sobhan Seyfaddini, The action spectrum and ${C}^0$ symplectic topology, arXiv preprint arXiv:1808.09790, 2018.
- Lev Buhovsky, Vincent Humilière, and Sobhan Seyfaddini, A $C^0$ counterexample to the Arnold conjecture, Invent. Math. 213 (2018), no. 2, 759–809. MR 3827210, DOI 10.1007/s00222-018-0797-x
- Lev Buhovsky and Yaron Ostrover, On the uniqueness of Hofer’s geometry, Geom. Funct. Anal. 21 (2011), no. 6, 1296–1330. MR 2860189, DOI 10.1007/s00039-011-0143-6
- Danny Calegari and Dongping Zhuang, Stable $W$-length, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 145–169. MR 2866929, DOI 10.1090/conm/560/11097
- Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308. MR 2476414, DOI 10.1090/S0273-0979-09-01249-X
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, and Steve Y. Oudot, Proximity of persistence modules and their diagrams, Proceedings of the twenty-fifth annual symposium on Computational geometry, ACM, 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
- Frédéric Chazal, Vin de Silva, and Steve Oudot, Persistence stability for geometric complexes, Geom. Dedicata 173 (2014), 193–214. MR 3275299, DOI 10.1007/s10711-013-9937-z
- Alexandru Oancea, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds [book review of MR3012475], Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 3, 521–530. MR 3362820, DOI 10.1090/S0273-0979-2015-01487-4
- Kai Cieliebak, Helmut Hofer, Janko Latschev, and Felix Schlenk, Quantitative symplectic geometry, Dynamics, ergodic theory, and geometry, Math. Sci. Res. Inst. Publ., vol. 54, Cambridge Univ. Press, Cambridge, 2007, pp. 1–44. MR 2369441, DOI 10.1017/CBO9780511755187.002
- 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
- David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko, Lipschitz functions have $L_p$-stable persistence, Found. Comput. Math. 10 (2010), no. 2, 127–139. MR 2594441, DOI 10.1007/s10208-010-9060-6
- William Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015), no. 5, 1550066, 8. MR 3323327, DOI 10.1142/S0219498815500668
- Vin de Silva and Robert Ghrist, Coverage in sensor networks via persistent homology, Algebr. Geom. Topol. 7 (2007), 339–358. MR 2308949, DOI 10.2140/agt.2007.7.339
- Vin de Silva and Vidit Nanda, Geometry in the space of persistence modules, Computational geometry (SoCG’13), ACM, New York, 2013, pp. 397–403. MR 3208238, DOI 10.1145/2462356.2462402
- Cornelia Druţu and Michael Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications, vol. 63, American Mathematical Society, Providence, RI, 2018. With an appendix by Bogdan Nica. MR 3753580, DOI 10.1090/coll/063
- Herbert Edelsbrunner, A short course in computational geometry and topology, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2014. MR 3328629, DOI 10.1007/978-3-319-05957-0
- Herbert Edelsbrunner and John Harer, Persistent homology—a survey, Surveys on discrete and computational geometry, Contemp. Math., vol. 453, Amer. Math. Soc., Providence, RI, 2008, pp. 257–282. MR 2405684, DOI 10.1090/conm/453/08802
- Herbert Edelsbrunner, David Letscher, and Afra Zomorodian, Topological persistence and simplification, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000) IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, pp. 454–463. MR 1931842, DOI 10.1109/SFCS.2000.892133
- Yakov Eliashberg, Sang Seon Kim, and Leonid Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol. 10 (2006), 1635–1747. MR 2284048, DOI 10.2140/gt.2006.10.1635
- Yakov Eliashberg and Leonid Polterovich, Bi-invariant metrics on the group of Hamiltonian diffeomorphisms, Internat. J. Math. 4 (1993), no. 5, 727–738. MR 1245350, DOI 10.1142/S0129167X93000352
- Michael Entov and Leonid Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 1635–1676. MR 1979584, DOI 10.1155/S1073792803210011
- Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770
- A. Floer and H. Hofer, Symplectic homology. I. Open sets in $\textbf {C}^n$, Math. Z. 215 (1994), no. 1, 37–88. MR 1254813, DOI 10.1007/BF02571699
- John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
- Robert Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 61–75. MR 2358377, DOI 10.1090/S0273-0979-07-01191-3
- Viktor L. Ginzburg, The Conley conjecture, Ann. of Math. (2) 172 (2010), no. 2, 1127–1180. MR 2680488, DOI 10.4007/annals.2010.172.1129
- Viktor L. Ginzburg and Başak Z. Gürel, Conley conjecture revisited, Int. Math. Res. Not. IMRN 3 (2019), 761–798. MR 3910472, DOI 10.1093/imrn/rnx137
- Eli Glasner and Benjamin Weiss, Topological groups with Rokhlin properties, Colloq. Math. 110 (2008), no. 1, 51–80. MR 2353899, DOI 10.4064/cm110-1-2
- M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- Jean Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds, J. Symplectic Geom. 15 (2017), no. 4, 1019–1069. MR 3734608, DOI 10.4310/JSG.2017.v15.n4.a3
- Jean Gutt and Michael Usher, Symplectically knotted codimension-zero embeddings of domains in $\Bbb {R}^4$, Duke Math. J. 168 (2019), no. 12, 2299–2363. MR 3999447, DOI 10.1215/00127094-2019-0013
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Jean-Claude Hausmann, On the Vietoris-Rips complexes and a cohomology theory for metric spaces, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 175–188. MR 1368659
- R. Hind and S. Lisi, Symplectic embeddings of polydisks, Selecta Math. (N.S.) 21 (2015), no. 3, 1099–1120. MR 3366927, DOI 10.1007/s00029-013-0146-2
- H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25–38. MR 1059642, DOI 10.1017/S0308210500024549
- H. Hofer and D. A. Salamon, Floer homology and Novikov rings, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 483–524. MR 1362838
- Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2011. Reprint of the 1994 edition. MR 2797558, DOI 10.1007/978-3-0348-0104-1
- Michael Hutchings, Beyond ECH capacities, Geom. Topol. 20 (2016), no. 2, 1085–1126. MR 3493100, DOI 10.2140/gt.2016.20.1085
- Nathan Jacobson, Basic algebra. II, W. H. Freeman and Co., San Francisco, Calif., 1980. MR 571884
- B. S. Kashin and S. V. Pastukhov, On short-term forecasting in the securities market, Dokl. Akad. Nauk 387 (2002), no. 6, 754–756 (Russian). MR 2006020
- Asaf Kislev and Egor Shelukhin, Bounds on spectral norms and barcodes, arXiv preprint arXiv: 1810.09865, 2018.
- A. S. Kronrod, On functions of two variables, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 1(35), 24–134 (Russian). MR 0034826
- François Lalonde and Dusa McDuff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), no. 2, 349–371. MR 1324138, DOI 10.2307/2118524
- Janko Latschev, Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold, Arch. Math. (Basel) 77 (2001), no. 6, 522–528. MR 1879057, DOI 10.1007/PL00000526
- Dorian Le Peutrec, Francis Nier, and Claude Viterbo, Precise Arrhenius law for $p$-forms: the Witten Laplacian and Morse-Barannikov complex, Ann. Henri Poincaré 14 (2013), no. 3, 567–610. MR 3035640, DOI 10.1007/s00023-012-0193-9
- Frédéric Le Roux, Sobhan Seyfaddini, and Claude Viterbo, Barcodes and area-preserving homeomorphisms, arXiv preprint arXiv:1810.03139, 2018.
- Michael Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math. 15 (2015), no. 3, 613–650. MR 3348168, DOI 10.1007/s10208-015-9255-y
- Michael W. Mahoney, Lek-Heng Lim, and Gunnar E. Carlsson, Algorithmic and statistical challenges in modern largescale data analysis are the focus of MMDS 2008, ACM SIGKDD Explorations Newsletter 10 (2008), no. 2, 57–60.
- Dan Mangoubi, On the inner radius of a nodal domain, Canad. Math. Bull. 51 (2008), no. 2, 249–260. MR 2414212, DOI 10.4153/CMB-2008-026-2
- Dusa McDuff, The Hofer conjecture on embedding symplectic ellipsoids, J. Differential Geom. 88 (2011), no. 3, 519–532. MR 2844441
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, 2nd ed., American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012. MR 2954391
- Anna Melnikov, $B$-orbits in solutions to the equation $X^2=0$ in triangular matrices, J. Algebra 223 (2000), no. 1, 101–108. MR 1738254, DOI 10.1006/jabr.1999.8056
- J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches, 1983) North-Holland, Amsterdam, 1984, pp. 1007–1057. MR 830252
- James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- Liviu Nicolaescu, An invitation to Morse theory, 2nd ed., Universitext, Springer, New York, 2011. MR 2883440, DOI 10.1007/978-1-4614-1105-5
- Partha Niyogi, Stephen Smale, and Shmuel Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39 (2008), no. 1-3, 419–441. MR 2383768, DOI 10.1007/s00454-008-9053-2
- Yong-Geun Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, The breadth of symplectic and Poisson geometry, Progr. Math., vol. 232, Birkhäuser Boston, Boston, MA, 2005, pp. 525–570. MR 2103018, DOI 10.1007/0-8176-4419-9_{1}8
- Yaron Ostrover, A comparison of Hofer’s metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Commun. Contemp. Math. 5 (2003), no. 5, 803–811. MR 2017719, DOI 10.1142/S0219199703001154
- Steve Y. Oudot, Persistence theory: from quiver representations to data analysis, Mathematical Surveys and Monographs, vol. 209, American Mathematical Society, Providence, RI, 2015. MR 3408277, DOI 10.1090/surv/209
- John Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol. 20 (2016), no. 2, 779–1034. MR 3493097, DOI 10.2140/gt.2016.20.779
- Iosif Polterovich, Leonid Polterovich, and Vukašin Stojisavljević, Persistence barcodes and Laplace eigenfunctions on surfaces, Geom. Dedicata 201 (2019), 111–138. MR 3978537, DOI 10.1007/s10711-018-0383-9
- Leonid Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13 (1993), no. 2, 357–367. MR 1235478, DOI 10.1017/S0143385700007410
- Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. MR 1826128, DOI 10.1007/978-3-0348-8299-6
- Leonid Polterovich, Inferring topology of quantum phase space, J. Appl. Comput. Topol. 2 (2018), no. 1-2, 61–82. With an appendix by Laurent Charles. MR 3873180, DOI 10.1007/s41468-018-0018-0
- Leonid Polterovich and Daniel Rosen, Function theory on symplectic manifolds, CRM Monograph Series, vol. 34, American Mathematical Society, Providence, RI, 2014. MR 3241729, DOI 10.1090/crmm/034
- 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
- Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević, Persistence modules with operators in Morse and Floer theory, Mosc. Math. J. 17 (2017), no. 4, 757–786. MR 3734662, DOI 10.17323/1609-4514-2016-16-4-757-786
- Leonid Polterovich and Mikhail Sodin, Nodal inequalities on surfaces, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 2, 459–467. MR 2364662, DOI 10.1017/S0305004107000175
- Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874, DOI 10.1016/0040-9383(93)90052-W
- M. Rudelson, Distances between non-symmetric convex bodies and the $MM^\ast$-estimate, Positivity 4 (2000), no. 2, 161–178. MR 1755679, DOI 10.1023/A:1009842406728
- Dietmar Salamon and Eduard Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), no. 10, 1303–1360. MR 1181727, DOI 10.1002/cpa.3160451004
- Felix Schlenk, Embedding problems in symplectic geometry, De Gruyter Expositions in Mathematics, vol. 40, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. MR 2147307, DOI 10.1515/9783110199697
- Matthias Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), no. 2, 419–461. MR 1755825, DOI 10.2140/pjm.2000.193.419
- Sobhan Seyfaddini, $C^0$-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants, Int. Math. Res. Not. IMRN 21 (2013), 4920–4960. MR 3123670, DOI 10.1093/imrn/rns191
- Egor Shelukhin, On the Hofer-Zehnder conjecture, arXiv preprint arXiv:1905.04769, 2019.
- L. A. Bunimovich, I. P. Cornfeld, R. L. Dobrushin, M. V. Jakobson, N. B. Maslova, Ya. B. Pesin, Ya. G. Sinaĭ, Yu. M. Sukhov, and A. M. Vershik, Dynamical systems. II, Encyclopaedia of Mathematical Sciences, vol. 2, Springer-Verlag, Berlin, 1989. Ergodic theory with applications to dynamical systems and statistical mechanics; Edited and with a preface by Sinaĭ; Translated from the Russian. MR 1024068
- A. I. Stepanets, Methods of approximation theory, VSP, Leiden, 2005. MR 2225312, DOI 10.1515/9783110195286
- Vukašin Stojisavljević and Jun Zhang, Persistence modules, symplectic Banach-Mazur distance and Riemannian metrics, arXiv preprint arXiv:1810.11151, 2018.
- Philip Thijsse, Upper triangular similarity of upper triangular matrices, Linear Algebra Appl. 260 (1997), 119–149. MR 1448353
- Michael Usher, Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, Israel J. Math. 184 (2011), 1–57. MR 2823968, DOI 10.1007/s11856-011-0058-9
- Michael Usher, Hofer’s metrics and boundary depth, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 57–128 (2013) (English, with English and French summaries). MR 3087390, DOI 10.24033/asens.2185
- Michael Usher, Symplectic Banach-Mazur distances between subsets of $\C ^n$, J. Topol. Anal., DOI: 10.1142/S179352532050048X.
- 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
- L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), no. 1, 454–472 (German). MR 1512371, DOI 10.1007/BF01447877
- Claude Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685–710. MR 1157321, DOI 10.1007/BF01444643
- Joa Weber, Noncontractible periodic orbits in cotangent bundles and Floer homology, Duke Math. J. 133 (2006), no. 3, 527–568. MR 2228462, DOI 10.1215/S0012-7094-06-13334-3
- Shmuel Weinberger, What is$\ldots$persistent homology?, Notices Amer. Math. Soc. 58 (2011), no. 1, 36–39. MR 2777589
- Shmuel Weinberger, Interpolation, the rudimentary geometry of spaces of Lipschitz functions, and geometric complexity, Found. Comput. Math. 19 (2019), no. 5, 991–1011. MR 4017679, DOI 10.1007/s10208-019-09416-0
- Eric W. Weisstein, Sum of squares function, http://mathworld.wolfram.com/SumofSquares Function.html.
- Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings, Topology 24 (1985), no. 2, 145–152. MR 793180, DOI 10.1016/0040-9383(85)90051-5
- Vladimir Alexandrovich Yudin, The multidimensional Jackson theorem, Mathematical notes of the Academy of Sciences of the USSR 20 (1976), no. 3, 801–804.
- Jun Zhang, $p$-cyclic persistent homology and Hofer distance, J. Symplectic Geom. 17 (2019), no. 3, 857–927. MR 4022215, DOI 10.4310/JSG.2019.v17.n3.a7