{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ZKEK3UGCBVCQJMM34FQYHDM523","short_pith_number":"pith:ZKEK3UGC","canonical_record":{"source":{"id":"1703.01392","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-04T03:44:35Z","cross_cats_sorted":["math.AT","math.DS"],"title_canon_sha256":"39112c5e1a2ebbb5c18a7dafed0bdd77e646bdb39c1a07f7bdf0345af1ed56ad","abstract_canon_sha256":"db5f604b231f9be5d73da7ca353873b60f2b036091c0ebf9bf6475c23fb51381"},"schema_version":"1.0"},"canonical_sha256":"ca88add0c20d4504b19be161838d9dd6da42d95c9790501e192f23fbb0420b9e","source":{"kind":"arxiv","id":"1703.01392","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.01392","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"arxiv_version","alias_value":"1703.01392v1","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.01392","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"pith_short_12","alias_value":"ZKEK3UGCBVCQ","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZKEK3UGCBVCQJMM3","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZKEK3UGC","created_at":"2026-05-18T12:31:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ZKEK3UGCBVCQJMM34FQYHDM523","target":"record","payload":{"canonical_record":{"source":{"id":"1703.01392","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-04T03:44:35Z","cross_cats_sorted":["math.AT","math.DS"],"title_canon_sha256":"39112c5e1a2ebbb5c18a7dafed0bdd77e646bdb39c1a07f7bdf0345af1ed56ad","abstract_canon_sha256":"db5f604b231f9be5d73da7ca353873b60f2b036091c0ebf9bf6475c23fb51381"},"schema_version":"1.0"},"canonical_sha256":"ca88add0c20d4504b19be161838d9dd6da42d95c9790501e192f23fbb0420b9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:30.901633Z","signature_b64":"YxStFJayM3GbQGmhcMZxgYkSCBF8THAzsxWSa4CXDaI95+y05o0Q8T+K6BuIZyxAhw91fJr2oRDT+tqAZ7myAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca88add0c20d4504b19be161838d9dd6da42d95c9790501e192f23fbb0420b9e","last_reissued_at":"2026-05-18T00:49:30.900856Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:30.900856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.01392","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:49:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AutFlDvbIOPZU4mUdnQhFvUJGvTJJAG6lOhzknXvrOBK5JzLCDjqeNh0FDF+EPHJF9KG6w9b++HugtfDMghnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T21:20:34.511566Z"},"content_sha256":"da5ee266ea83e0d916d57a5f8ec4e440cf1e0df4d4258d358f65a727fcf45f29","schema_version":"1.0","event_id":"sha256:da5ee266ea83e0d916d57a5f8ec4e440cf1e0df4d4258d358f65a727fcf45f29"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ZKEK3UGCBVCQJMM34FQYHDM523","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Persistence modules with operators in Morse and Floer theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.DS"],"primary_cat":"math.SG","authors_text":"Egor Shelukhin, Leonid Polterovich, Vuka\\v{s}in Stojisavljevi\\'c","submitted_at":"2017-03-04T03:44:35Z","abstract_excerpt":"We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01392","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:49:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BIfIhKHk8FQ+5/HUeptHDYy2+Nc7vGnPBPe4nHaiIy40GT5B7ZxAj8Y/AO73p8pruZQHk9l12BvG61Mubl4FBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T21:20:34.512305Z"},"content_sha256":"f983f949b1afdbb849621b1d2337b17bc2476aece7592c7ebcb6a66631c6ead0","schema_version":"1.0","event_id":"sha256:f983f949b1afdbb849621b1d2337b17bc2476aece7592c7ebcb6a66631c6ead0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZKEK3UGCBVCQJMM34FQYHDM523/bundle.json","state_url":"https://pith.science/pith/ZKEK3UGCBVCQJMM34FQYHDM523/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZKEK3UGCBVCQJMM34FQYHDM523/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T21:20:34Z","links":{"resolver":"https://pith.science/pith/ZKEK3UGCBVCQJMM34FQYHDM523","bundle":"https://pith.science/pith/ZKEK3UGCBVCQJMM34FQYHDM523/bundle.json","state":"https://pith.science/pith/ZKEK3UGCBVCQJMM34FQYHDM523/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZKEK3UGCBVCQJMM34FQYHDM523/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZKEK3UGCBVCQJMM34FQYHDM523","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"db5f604b231f9be5d73da7ca353873b60f2b036091c0ebf9bf6475c23fb51381","cross_cats_sorted":["math.AT","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-04T03:44:35Z","title_canon_sha256":"39112c5e1a2ebbb5c18a7dafed0bdd77e646bdb39c1a07f7bdf0345af1ed56ad"},"schema_version":"1.0","source":{"id":"1703.01392","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.01392","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"arxiv_version","alias_value":"1703.01392v1","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.01392","created_at":"2026-05-18T00:49:30Z"},{"alias_kind":"pith_short_12","alias_value":"ZKEK3UGCBVCQ","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZKEK3UGCBVCQJMM3","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZKEK3UGC","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:f983f949b1afdbb849621b1d2337b17bc2476aece7592c7ebcb6a66631c6ead0","target":"graph","created_at":"2026-05-18T00:49:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.","authors_text":"Egor Shelukhin, Leonid Polterovich, Vuka\\v{s}in Stojisavljevi\\'c","cross_cats":["math.AT","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-04T03:44:35Z","title":"Persistence modules with operators in Morse and Floer theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01392","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:da5ee266ea83e0d916d57a5f8ec4e440cf1e0df4d4258d358f65a727fcf45f29","target":"record","created_at":"2026-05-18T00:49:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"db5f604b231f9be5d73da7ca353873b60f2b036091c0ebf9bf6475c23fb51381","cross_cats_sorted":["math.AT","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-04T03:44:35Z","title_canon_sha256":"39112c5e1a2ebbb5c18a7dafed0bdd77e646bdb39c1a07f7bdf0345af1ed56ad"},"schema_version":"1.0","source":{"id":"1703.01392","kind":"arxiv","version":1}},"canonical_sha256":"ca88add0c20d4504b19be161838d9dd6da42d95c9790501e192f23fbb0420b9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca88add0c20d4504b19be161838d9dd6da42d95c9790501e192f23fbb0420b9e","first_computed_at":"2026-05-18T00:49:30.900856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:30.900856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YxStFJayM3GbQGmhcMZxgYkSCBF8THAzsxWSa4CXDaI95+y05o0Q8T+K6BuIZyxAhw91fJr2oRDT+tqAZ7myAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:30.901633Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.01392","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da5ee266ea83e0d916d57a5f8ec4e440cf1e0df4d4258d358f65a727fcf45f29","sha256:f983f949b1afdbb849621b1d2337b17bc2476aece7592c7ebcb6a66631c6ead0"],"state_sha256":"fa0f11a3fc4daa926b4ee18711b13cc658959c0f77e498b3a605d015a1a968ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9bla/ikiFHY3n6cYi7Py9fNsjA7Oh98AATJ33O5H8Ampg452T8v7U8M6HS0BYJ5N1XZsNu8iVqRHxG4L+HrfBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T21:20:34.516139Z","bundle_sha256":"37aead3a9923c4ead713dfd81ddc938b381ff0b67ea61b7287771555568ae274"}}