Volume 25, issue 7 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Bounds on spectral norms and barcodes

Asaf Kislev and Egor Shelukhin

Geometry & Topology 25 (2021) 3257–3350
Bibliography
1 M Abouzaid, Symplectic cohomology and Viterbo’s theorem, from: "Free loop spaces in geometry and topology" (editors J Latschev, A Oancea), IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc. (2015) 271 MR3444367
2 M Abouzaid, T Kragh, Simple homotopy equivalence of nearby Lagrangians, Acta Math. 220 (2018) 207 MR3849284
3 M Abouzaid, P Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010) 627 MR2602848
4 M Abouzaid, I Smith, Khovanov homology from Floer cohomology, J. Amer. Math. Soc. 32 (2019) 1 MR3867999
5 L Abrams, The quantum Euler class and the quantum cohomology of the Grassmannians, Israel J. Math. 117 (2000) 335 MR1760598
6 P Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. 2008 (2008) MR2424172
7 D Alvarez-Gavela, V Kaminker, A Kislev, K Kliakhandler, A Pavlichenko, L Rigolli, D Rosen, O Shabtai, B Stevenson, J Zhang, Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms, J. Topol. Anal. 11 (2019) 467 MR3958929
8 M Audin, On the topology of Lagrangian submanifolds : examples and counter-examples, Port. Math. 62 (2005) 375 MR2191628
9 M Audin, Lagrangian skeletons, periodic geodesic flows and symplectic cuttings, Manuscripta Math. 124 (2007) 533 MR2357797
10 D Auroux, A beginner’s introduction to Fukaya categories, from: "Contact and symplectic topology" (editors F Bourgeois, V Colin, A Stipsicz), Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc. (2014) 85 MR3220941
11 S A Barannikov, The framed Morse complex and its invariants, from: "Singularities and bifurcations" (editor V I Arnold), Adv. Soviet Math. 21, Amer. Math. Soc. (1994) 93 MR1310596
12 J F Barraud, O Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. 166 (2007) 657 MR2373371
13 U Bauer, M Lesnick, Induced matchings and the algebraic stability of persistence barcodes, J. Comput. Geom. 6 (2015) 162 MR3333456
14 A L Besse, Manifolds all of whose geodesics are closed, 93, Springer (1978) MR496885
15 P Biran, Lagrangian non-intersections, Geom. Funct. Anal. 16 (2006) 279 MR2231465
16 P Biran, O Cornea, Quantum structures for Lagrangian submanifolds, preprint (2007) arXiv:0708.4221
17 P Biran, O Cornea, A Lagrangian quantum homology, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 1 MR2555932
18 P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881 MR2546618
19 P Biran, O Cornea, E Shelukhin, Lagrangian shadows and triangulated categories, preprint (2018) arXiv:1806.06630
20 P Biran, C Membrez, The Lagrangian cubic equation, Int. Math. Res. Not. 2016 (2016) 2569 MR3519124
21 L Buhovsky, V Humilière, S Seyfaddini, The action spectrum and C0 symplectic topology, Math. Ann. 380 (2021) 293 MR4263685
22 D Calegari, scl, 20, Mathematical Society of Japan (2009) MR2527432
23 G Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009) 255 MR2476414
24 G. Carlsson, A. Zomorodian, A. Collins, L Guibas, Persistence barcodes for shapes, Int. J. Shape Modeling 11 (2005) 149
25 F Charest, C Woodward, Fukaya algebras via stabilizing divisors, preprint (2015) arXiv:1505.08146
26 F Charest, C Woodward, Floer trajectories and stabilizing divisors, J. Fixed Point Theory Appl. 19 (2017) 1165 MR3659006
27 F Charette, A geometric refinement of a theorem of Chekanov, J. Symplectic Geom. 10 (2012) 475 MR2983439
28 F Charette, O Cornea, Categorification of Seidel’s representation, Israel J. Math. 211 (2016) 67 MR3474957
29 F Chazal, V de Silva, M Glisse, S Oudot, The structure and stability of persistence modules, Springer (2016) MR3524869
30 Y V Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J. 95 (1998) 213 MR1646550
31 Y V Chekanov, Invariant Finsler metrics on the space of Lagrangian embeddings, Math. Z. 234 (2000) 605 MR1774099
32 D Cohen-Steiner, H Edelsbrunner, J Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007) 103 MR2279866
33 O Cornea, A Ranicki, Rigidity and gluing for Morse and Novikov complexes, J. Eur. Math. Soc. 5 (2003) 343 MR2017851
34 O Cornea, E Shelukhin, Lagrangian cobordism and metric invariants, J. Differential Geom. 112 (2019) 1 MR3948226
35 W Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015) MR3323327
36 H Edelsbrunner, D Letscher, A Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002) 511 MR1949898
37 M Entov, Quasi-morphisms and quasi-states in symplectic topology, from: "Proceedings of the International Congress of Mathematicians" (editors S Y Jang, Y R Kim, D W Lee, I Ye), Kyung Moon Sa (2014) 1147 MR3728656
38 M Entov, L Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003) 1635 MR1979584
39 M Entov, L Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009) 773 MR2507748
40 A Floer, Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math. 42 (1989) 335 MR990135
41 M Fraser, Contact spectral invariants and persistence, preprint (2015) arXiv:1502.05979
42 U Frauenfelder, F Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math. 159 (2007) 1 MR2342472
43 K Fukaya, Unobstructed immersed Lagrangian correspondence and filtered A functor, preprint (2017) arXiv:1706.02131
44 K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, I, 46, Amer. Math. Soc. (2009) MR2553465
45 K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, II, 46, Amer. Math. Soc. (2009) MR2548482
46 K Fukaya, Y G Oh, H Ohta, K Ono, Displacement of polydisks and Lagrangian Floer theory, J. Symplectic Geom. 11 (2013) 231 MR3046491
47 K Fukaya, Y G Oh, H Ohta, K Ono, Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory, 1254, Amer. Math. Soc. (2019) MR3986938
48 K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933 MR1688434
49 S Ganatra, J Pardon, V Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci. 131 (2020) 73 MR4106794
50 R Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. 45 (2008) 61 MR2358377
51 V L Ginzburg, B Z Gürel, Hamiltonian pseudo-rotations of projective spaces, Invent. Math. 214 (2018) 1081 MR3878727
52 P Griffiths, J Morgan, Rational homotopy theory and differential forms, 16, Springer (2013) MR3136262
53 H Hofer, Lusternik–Schnirelman-theory for Lagrangian intersections, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988) 465 MR970850
54 H Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 25 MR1059642
55 H H W Hofer, Polyfolds and Fredholm theory, from: "Lectures on geometry" (editor N M J Woodhouse), Oxford Univ. Press (2017) 87 MR3676594
56 H Hofer, K Wysocki, E Zehnder, sc-smoothness, retractions and new models for smooth spaces, Discrete Contin. Dyn. Syst. 28 (2010) 665 MR2644764
57 H Hofer, K Wysocki, E Zehnder, Applications of polyfold theory, I : The polyfolds of Gromov–Witten theory, 1179, Amer. Math. Soc. (2017) MR3683060
58 H Hofer, K Wysocki, E Zehnder, Polyfold and Fredholm theory, 72, Springer (2021) 1001 MR4298268
59 S Hu, F Lalonde, A relative Seidel morphism and the Albers map, Trans. Amer. Math. Soc. 362 (2010) 1135 MR2563724
60 S Hu, F Lalonde, R Leclercq, Homological Lagrangian monodromy, Geom. Topol. 15 (2011) 1617 MR2851073
61 V Humilière, R Leclercq, S Seyfaddini, Coisotropic rigidity and C0–symplectic geometry, Duke Math. J. 164 (2015) 767 MR3322310
62 C Hyvrier, Lagrangian circle actions, Algebr. Geom. Topol. 16 (2016) 1309 MR3523040
63 J Katić, D Milinković, Piunikhin–Salamon–Schwarz isomorphisms for Lagrangian intersections, Differential Geom. Appl. 22 (2005) 215 MR2122744
64 T Kimura, J Stasheff, A A Voronov, On operad structures of moduli spaces and string theory, Comm. Math. Phys. 171 (1995) 1 MR1341693
65 F Lalonde, D McDuff, The geometry of symplectic energy, Ann. of Math. 141 (1995) 349 MR1324138
66 F Le Roux, S Seyfaddini, C Viterbo, Barcodes and area-preserving homeomorphisms, Geom. Topol. 25 (2021) 2713 MR4347306
67 R Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn. 2 (2008) 249 MR2383268
68 R Leclercq, F Zapolsky, Spectral invariants for monotone Lagrangians, J. Topol. Anal. 10 (2018) 627 MR3850107
69 Y Lekili, J Pascaleff, Floer cohomology of 𝔤–equivariant Lagrangian branes, Compos. Math. 152 (2016) 1071 MR3505649
70 E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247 MR1338784
71 S Lisi, A Rieser, Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings, J. Symplectic Geom. 18 (2020) 819 MR4142489
72 C C M Liu, Moduli of J–holomorphic curves with Lagrangian boundary conditions and open Gromov–Witten invariants for an S1–equivariant pair, preprint (2002) arXiv:math/0210257
73 G Liu, G Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998) 1 MR1642105
74 G Lu, Gromov–Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1 MR2282367
75 D McDuff, Constructing the virtual fundamental class of a Kuranishi atlas, Algebr. Geom. Topol. 19 (2019) 151 MR3910580
76 D McDuff, L Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994) 405 MR1262938
77 D McDuff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2012) MR2954391
78 D McDuff, D Salamon, Introduction to symplectic topology, Oxford Univ. Press (2017) MR3674984
79 D McDuff, S Tolman, Topological properties of Hamiltonian circle actions, Int. Math. Res. Pap. (2006) MR2210662
80 A Monzner, N Vichery, F Zapolsky, Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization, J. Mod. Dyn. 6 (2012) 205 MR2968955
81 A Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology, from: "Symplectic geometry and Floer homology : a survey of the Floer homology for manifolds with contact type boundary or symplectic homology", Ensaios Mat. 7, Soc. Brasil. Mat. (2004) 51 MR2100955
82 Y G Oh, Riemann–Hilbert problem and application to the perturbation theory of analytic discs, Kyungpook Math. J. 35 (1995) 39 MR1345070
83 Y G Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Int. Math. Res. Not. 1996 (1996) 305 MR1389956
84 Y G Oh, Symplectic topology as the geometry of action functional, II : Pants product and cohomological invariants, Comm. Anal. Geom. 7 (1999) 1 MR1674121
85 Y G Oh, Floer homology and its continuity for non-compact Lagrangian submanifolds, Turkish J. Math. 25 (2001) 103 MR1829082
86 Y G Oh, Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002) 579 MR1958084
87 Y G Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, from: "The breadth of symplectic and Poisson geometry" (editors J E Marsden, T S Ratiu), Progr. Math. 232, Birkhäuser (2005) 525 MR2103018
88 Y G Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Math. J. 130 (2005) 199 MR2181090
89 Y G Oh, Symplectic topology and Floer homology, I : Symplectic geometry and pseudoholomorphic curves, 28, Cambridge Univ. Press (2015) MR3443239
90 Y G Oh, Symplectic topology and Floer homology, II : Floer homology and its applications, 29, Cambridge Univ. Press (2015) MR3524783
91 Y G Oh, K Zhu, Floer trajectories with immersed nodes and scale-dependent gluing, J. Symplectic Geom. 9 (2011) 483 MR2900788
92 J Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol. 20 (2016) 779 MR3493097
93 S Piunikhin, D Salamon, M Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, from: "Contact and symplectic geometry" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171 MR1432464
94 L Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13 (1993) 357 MR1235478
95 L Polterovich, The geometry of the group of symplectic diffeomorphisms, Birkhäuser (2001) MR1826128
96 L Polterovich, D Rosen, Function theory on symplectic manifolds, 34, Amer. Math. Soc. (2014) MR3241729
97 L Polterovich, E Shelukhin, Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules, Selecta Math. 22 (2016) 227 MR3437837
98 L Polterovich, E Shelukhin, V Stojisavljević, Persistence modules with operators in Morse and Floer theory, Mosc. Math. J. 17 (2017) 757 MR3734662
99 J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827 MR1241874
100 M Schwarz, A quantum cup-length estimate for symplectic fixed points, Invent. Math. 133 (1998) 353 MR1632778
101 M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419 MR1755825
102 P Seidel, π1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046 MR1487754
103 P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103 MR1765826
104 P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc. (2008) MR2441780
105 P Seidel, Disjoinable Lagrangian spheres and dilations, Invent. Math. 197 (2014) 299 MR3232008
106 S Seyfaddini, C0–limits of Hamiltonian paths and the Oh–Schwarz spectral invariants, Int. Math. Res. Not. 2013 (2013) 4920 MR3123670
107 E Shelukhin, Viterbo conjecture for Zoll symmetric spaces, preprint (2018) arXiv:1811.05552
108 E Shelukhin, On the Hofer–Zehnder conjecture, preprint (2019) arXiv:1905.04769
109 E Shelukhin, D Tonkonog, R Vianna, Geometry of symplectic flux and Lagrangian torus fibrations, preprint (2018) arXiv:1804.02044
110 V de Silva, J W Robbin, D A Salamon, Combinatorial Floer homology, 1080, Amer. Math. Soc. (2014) MR3205426
111 I Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012) 149 MR2929086
112 N Steenrod, The topology of fibre bundles, 14, Princeton Univ. Press (1951) MR0039258
113 B Stevenson, A quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric, Israel J. Math. 223 (2018) 141 MR3773059
114 M Usher, Spectral numbers in Floer theories, Compos. Math. 144 (2008) 1581 MR2474322
115 M Usher, The sharp energy-capacity inequality, Commun. Contemp. Math. 12 (2010) 457 MR2661273
116 M Usher, Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, Israel J. Math. 184 (2011) 1 MR2823968
117 M Usher, Hofer’s metrics and boundary depth, Ann. Sci. Éc. Norm. Supér. 46 (2013) 57 MR3087390
118 M Usher, Observations on the Hofer distance between closed subsets, Math. Res. Lett. 22 (2015) 1805 MR3507263
119 M Usher, J Zhang, Persistent homology and Floer–Novikov theory, Geom. Topol. 20 (2016) 3333 MR3590354
120 C Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens, Bull. Soc. Math. France 115 (1987) 361 MR926533
121 C Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992) 685 MR1157321
122 F Zapolsky, The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory, preprint (2015) arXiv:1507.02253
123 J Zhang, p–cyclic persistent homology and Hofer distance, J. Symplectic Geom. 17 (2019) 857 MR4022215
124 A Zomorodian, G Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005) 249 MR2121296