{"id":"https://openalex.org/W4384944304","doi":"https://doi.org/10.1007/s00454-022-00461-1","title":"Discrete-to-Continuous Extensions: Lov\u00e1sz Extension and Morse Theory","display_name":"Discrete-to-Continuous Extensions: Lov\u00e1sz Extension and Morse Theory","publication_year":2023,"publication_date":"2023-07-21","ids":{"openalex":"https://openalex.org/W4384944304","doi":"https://doi.org/10.1007/s00454-022-00461-1"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-022-00461-1","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-022-00461-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-022-00461-1.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-022-00461-1.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072990982","display_name":"J\u00fcrgen Jost","orcid":"https://orcid.org/0000-0001-5258-6590"},"institutions":[{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]},{"id":"https://openalex.org/I4210119712","display_name":"Max Planck Institute for Mathematics","ror":"https://ror.org/02dh8ja68","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210119712"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"J\u00fcrgen Jost","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany","Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327","https://openalex.org/I4210119712"]},{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100366127","display_name":"Dong Zhang","orcid":"https://orcid.org/0000-0002-8439-9156"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]},{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]},{"id":"https://openalex.org/I4210119712","display_name":"Max Planck Institute for Mathematics","ror":"https://ror.org/02dh8ja68","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210119712"]}],"countries":["CN","DE"],"is_corresponding":true,"raw_author_name":"Dong Zhang","raw_affiliation_strings":["LMAM and School of Mathematical Sciences, Peking University, Beijing, China","Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany","Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany"],"raw_orcid":"https://orcid.org/0000-0002-8439-9156","affiliations":[{"raw_affiliation_string":"LMAM and School of Mathematical Sciences, Peking University, Beijing, China","institution_ids":["https://openalex.org/I20231570"]},{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327","https://openalex.org/I4210119712"]},{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5072990982","https://openalex.org/A5100366127"],"corresponding_institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210091327","https://openalex.org/I4210119712"],"apc_list":{"value":3090,"currency":"USD","value_usd":3090},"apc_paid":{"value":3090,"currency":"USD","value_usd":3090},"fwci":0.2802,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.56997059,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"72","issue":"1","first_page":"49","last_page":"72"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9793999791145325,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10086","display_name":"Alzheimer's disease research and treatments","score":0.9136999845504761,"subfield":{"id":"https://openalex.org/subfields/2737","display_name":"Physiology"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discrete-morse-theory","display_name":"Discrete Morse theory","score":0.9448078870773315},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.772282600402832},{"id":"https://openalex.org/keywords/morse-code","display_name":"Morse code","score":0.7339841723442078},{"id":"https://openalex.org/keywords/circle-valued-morse-theory","display_name":"Circle-valued Morse theory","score":0.708789050579071},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.6099730134010315},{"id":"https://openalex.org/keywords/morse-theory","display_name":"Morse theory","score":0.5729423761367798},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5296401977539062},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"Simplicial complex","score":0.5215599536895752},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.44771116971969604},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4038517475128174},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37555214762687683},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1367362141609192},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12449422478675842}],"concepts":[{"id":"https://openalex.org/C170315498","wikidata":"https://www.wikidata.org/wiki/Q5282039","display_name":"Discrete Morse theory","level":3,"score":0.9448078870773315},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.772282600402832},{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.7339841723442078},{"id":"https://openalex.org/C24296071","wikidata":"https://www.wikidata.org/wiki/Q5121372","display_name":"Circle-valued Morse theory","level":3,"score":0.708789050579071},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.6099730134010315},{"id":"https://openalex.org/C156730297","wikidata":"https://www.wikidata.org/wiki/Q2296951","display_name":"Morse theory","level":2,"score":0.5729423761367798},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5296401977539062},{"id":"https://openalex.org/C187929450","wikidata":"https://www.wikidata.org/wiki/Q994399","display_name":"Simplicial complex","level":2,"score":0.5215599536895752},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44771116971969604},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4038517475128174},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37555214762687683},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1367362141609192},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12449422478675842},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00454-022-00461-1","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-022-00461-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-022-00461-1.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00454-022-00461-1","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00454-022-00461-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-022-00461-1.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320327897","display_name":"Max-Planck-Institut f\u00fcr Mathematik in den Naturwissenschaften","ror":"https://ror.org/00ez2he07"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4384944304.pdf","grobid_xml":"https://content.openalex.org/works/W4384944304.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W23324085","https://openalex.org/W317369104","https://openalex.org/W1498671329","https://openalex.org/W1540633728","https://openalex.org/W1548333856","https://openalex.org/W1570807537","https://openalex.org/W1940289637","https://openalex.org/W2005179357","https://openalex.org/W2013083786","https://openalex.org/W2013451459","https://openalex.org/W2014823407","https://openalex.org/W2027984435","https://openalex.org/W2033810643","https://openalex.org/W2037613900","https://openalex.org/W2064020260","https://openalex.org/W2068229848","https://openalex.org/W2082598343","https://openalex.org/W2109978147","https://openalex.org/W2133227973","https://openalex.org/W2146019744","https://openalex.org/W2165125983","https://openalex.org/W2322759512","https://openalex.org/W2486409092","https://openalex.org/W2782160304","https://openalex.org/W2908292748","https://openalex.org/W2952019053","https://openalex.org/W2962683434","https://openalex.org/W2963081029","https://openalex.org/W2963210866","https://openalex.org/W2964961059","https://openalex.org/W3099422512","https://openalex.org/W3140579943","https://openalex.org/W4206054140","https://openalex.org/W4226412082","https://openalex.org/W4247131145","https://openalex.org/W4293713108","https://openalex.org/W4300596011","https://openalex.org/W6603944243"],"related_works":["https://openalex.org/W4389768470","https://openalex.org/W1595332435","https://openalex.org/W2037435271","https://openalex.org/W2185671344","https://openalex.org/W2106662121","https://openalex.org/W1968935960","https://openalex.org/W2548670722","https://openalex.org/W4308897745","https://openalex.org/W4281700539","https://openalex.org/W1966769279"],"abstract_inverted_index":{"Abstract":[0],"This":[1],"is":[2],"the":[3,21,42,47,56,61,71,80,118],"first":[4],"of":[5,8,49,70,120],"a":[6,12,38,67,96],"series":[7],"papers":[9],"that":[10],"develop":[11],"systematic":[13],"bridge":[14,79],"between":[15,32],"constructions":[16],"in":[17],"discrete":[18,34,86,97],"mathematics":[19],"and":[20,41,84,108],"corresponding":[22],"continuous":[23,43],"analogs.":[24],"In":[25],"this":[26],"paper,":[27],"we":[28,65,93],"establish":[29],"an":[30],"equivalence":[31],"Forman\u2019s":[33],"Morse":[35,44,53,98,106],"theory":[36,45,99,107],"on":[37,55,74,88,100],"simplicial":[39,76],"complex":[40,59],"(in":[46],"sense":[48],"any":[50],"known":[51],"non-smooth":[52],"theory)":[54],"associated":[57],"order":[58],"via":[60],"Lov\u00e1sz":[62,109],"extension.":[63],"Furthermore,":[64],"propose":[66],"new":[68,114],"version":[69],"Lusternik\u2013Schnirelman":[72,82],"category":[73],"abstract":[75],"complexes":[77],"to":[78,112],"classical":[81],"theorem":[83],"its":[85],"analog":[87],"finite":[89],"complexes.":[90],"More":[91],"generally,":[92],"can":[94],"suggest":[95],"hypergraphs":[101],"by":[102],"employing":[103],"piecewise-linear":[104],"(PL)":[105],"extension,":[110],"hoping":[111],"provide":[113],"tools":[115],"for":[116],"exploring":[117],"structure":[119],"hypergraphs.":[121]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}