{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20395"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20395","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Hanna Neumann conjecture: A flow detection approach","abstract":"\"The Hanna Neumann Conjecture states that if two subgroups of a finitely generated free group have finite ranks m and n, then their intersection has rank N which satisfies $N$ $-$ 1 $\\leq$ ($m$ $-$ 1)($n$ $-$ 1). The current work examines this conjecture by restating it in terms of a stronger conjecture on the pullback in a particular category of graphs. The notion of a vertex pairing is developed, and shown to have a direct bearing on the pullback conjecture. In this light, the Flow Conjecture is stated, and its relationship as an apparently stronger conjecture than the Hanna Neumann Conjecture becomes evident. We then prove the Flow Conjecture for certain special cases by detecting a special kind of \"\"short\"\" flow. Finally, we examine the properties of projection and non-projection flows that are intended to lead to a full solution to the Hanna Neumann Conjecture.\"","abstract_html":""The Hanna Neumann Conjecture states that if two subgroups of a finitely generated free group have finite ranks m and n, then their intersection has rank N which satisfies $N$ $-$ 1 $\\leq$ ($m$ $-$ 1)($n$ $-$ 1). The current work examines this conjecture by restating it in terms of a stronger conjecture on the pullback in a particular category of graphs. The notion of a vertex pairing is developed, and shown to have a direct bearing on the pullback conjecture. In this light, the Flow Conjecture is stated, and its relationship as an apparently stronger conjecture than the Hanna Neumann Conjecture becomes evident. We then prove the Flow Conjecture for certain special cases by detecting a special kind of ""short"" flow. Finally, we examine the properties of projection and non-projection flows that are intended to lead to a full solution to the Hanna Neumann Conjecture."","abstract_has_math":true,"creators":["Feuerman, Kenneth Edward"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:37:58Z","date_published":"2011-05-07T12:37:58Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1991 Feuerman, Kenneth Edward"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9124411","(UMI)AAI9124411"],"render_values":[{"text":"AAI9124411","href":null,"code":true},{"text":"(UMI)AAI9124411","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20395","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Feuerman, Kenneth Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:37:58Z","10000-01-01","1991"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1991 Feuerman, Kenneth Edward"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9124411","(UMI)AAI9124411","http://hdl.handle.net/2142/20395"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"The Hanna Neumann Conjecture states that if two subgroups of a finitely generated free group have finite ranks m and n, then their intersection has rank N which satisfies $N$ $-$ 1 $\\leq$ ($m$ $-$ 1)($n$ $-$ 1). The current work examines this conjecture by restating it in terms of a stronger conjecture on the pullback in a particular category of graphs. The notion of a vertex pairing is developed, and shown to have a direct bearing on the pullback conjecture. In this light, the Flow Conjecture is stated, and its relationship as an apparently stronger conjecture than the Hanna Neumann Conjecture becomes evident. We then prove the Flow Conjecture for certain special cases by detecting a special kind of \"\"short\"\" flow. Finally, we examine the properties of projection and non-projection flows that are intended to lead to a full solution to the Hanna Neumann Conjecture.\"","Made available in DSpace on 2011-05-07T12:37:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9124411.pdf: 3726293 bytes, checksum: 74090a5c0abded7f33602ae983d19f02 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:36Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:06-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["The Hanna Neumann conjecture: A flow detection approach"]}]}],"canonical_facts":{"dc:creator":["Feuerman, Kenneth Edward"],"dc:date":["2011-05-07T12:37:58Z","10000-01-01","1991"],"dc:description":["\"The Hanna Neumann Conjecture states that if two subgroups of a finitely generated free group have finite ranks m and n, then their intersection has rank N which satisfies $N$ $-$ 1 $\\leq$ ($m$ $-$ 1)($n$ $-$ 1). The current work examines this conjecture by restating it in terms of a stronger conjecture on the pullback in a particular category of graphs. The notion of a vertex pairing is developed, and shown to have a direct bearing on the pullback conjecture. In this light, the Flow Conjecture is stated, and its relationship as an apparently stronger conjecture than the Hanna Neumann Conjecture becomes evident. We then prove the Flow Conjecture for certain special cases by detecting a special kind of \"\"short\"\" flow. Finally, we examine the properties of projection and non-projection flows that are intended to lead to a full solution to the Hanna Neumann Conjecture.\"","Made available in DSpace on 2011-05-07T12:37:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9124411.pdf: 3726293 bytes, checksum: 74090a5c0abded7f33602ae983d19f02 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:36Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:06-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9124411","(UMI)AAI9124411","http://hdl.handle.net/2142/20395"],"dc:language":["eng"],"dc:rights":["Copyright 1991 Feuerman, Kenneth Edward"],"dc:subject":["Mathematics"],"dc:title":["The Hanna Neumann conjecture: A flow detection approach"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}