{"id":{"repo_id":"unr","oai_identifier":"oai:scholarwolf.unr.edu:11714/7503"},"canonical_url":"https://search.dev.ndltd.org/etd/unr/oai:scholarwolf.unr.edu:11714/7503","repository":{"repo_id":"unr","name":"University of Nevada - Reno","base_url":"https://scholarwolf.unr.edu/server/oai/request"},"display":{"title":"The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants","abstract":"We review a theorem of G.V. Belyi which establishes an equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: finite bipartite graphs embedded on smooth, oriented, compact topological surfaces. Through these categorical equivalences, one obtains a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ''Zapponi orientability\" and ''twist-orient type\". We conclude with speculations for future work on ''Zapponi orientability\" and the Grothendieck-Teichmuller group.","abstract_html":"We review a theorem of G.V. Belyi which establishes an equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: finite bipartite graphs embedded on smooth, oriented, compact topological surfaces. Through these categorical equivalences, one obtains a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ''Zapponi orientability" and ''twist-orient type". We conclude with speculations for future work on ''Zapponi orientability" and the Grothendieck-Teichmuller group.","abstract_has_math":false,"creators":["Mallory, Daniel"],"institution":"University of Nevada, Reno","degree_name":"Mathematics","degree_level":"Honors Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Rogers, Christopher L."],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-05-01","date_published":"2020-05-01","updated_at":"2026-07-27T21:46:41Z","subjects":[],"languages":["en_US","English"],"rights":["Creative Commons Attribution 4.0 United States"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11714/7503","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Rogers, Christopher L."]},{"key":"dc:creator","label":"Author","values":["Mallory, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-07-21T16:46:01Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-07-21T16:46:01Z"]},{"key":"dc:date.issued","label":"Date","values":["2020-05-01"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Honors Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Nevada, Reno"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["Creative Commons Attribution 4.0 United States"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11714/7503"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The University of Nevada, Reno Libraries will promptly respond to removal requests related to content that violates intellectual property laws, data protections, or has been uploaded without creator consent. Takedown notices should be directed to our ScholarWolf team (scholarwolf@library.unr.edu) with information about the object, including its full URL and the nature of your complaint."]},{"key":"dc:description.abstract","label":"Abstract","values":["We review a theorem of G.V. Belyi which establishes an equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: finite bipartite graphs embedded on smooth, oriented, compact topological surfaces. Through these categorical equivalences, one obtains a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ''Zapponi orientability\" and ''twist-orient type\". We conclude with speculations for future work on ''Zapponi orientability\" and the Grothendieck-Teichmuller group."]},{"key":"dc:format","label":"Dc Format","values":["PDF"]},{"key":"dc:title","label":"Title","values":["The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants"]}]}],"canonical_facts":{"dc:contributor.advisor":["Rogers, Christopher L."],"dc:creator":["Mallory, Daniel"],"dc:date.accessioned":["2020-07-21T16:46:01Z"],"dc:date.available":["2020-07-21T16:46:01Z"],"dc:date.issued":["2020-05-01"],"dc:description":["The University of Nevada, Reno Libraries will promptly respond to removal requests related to content that violates intellectual property laws, data protections, or has been uploaded without creator consent. Takedown notices should be directed to our ScholarWolf team (scholarwolf@library.unr.edu) with information about the object, including its full URL and the nature of your complaint."],"dc:description.abstract":["We review a theorem of G.V. Belyi which establishes an equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: finite bipartite graphs embedded on smooth, oriented, compact topological surfaces. Through these categorical equivalences, one obtains a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ''Zapponi orientability\" and ''twist-orient type\". We conclude with speculations for future work on ''Zapponi orientability\" and the Grothendieck-Teichmuller group."],"dc:format":["PDF"],"dc:identifier.uri":["http://hdl.handle.net/11714/7503"],"dc:language":["English"],"dc:language.iso":["en_US"],"dc:rights":["Creative Commons Attribution 4.0 United States"],"dc:title":["The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants"],"dc:type":["Thesis"],"thesis:degree_level":["Honors Thesis"],"thesis:degree_name":["Mathematics"],"thesis:institution_name":["University of Nevada, Reno"]},"updated_at":"2026-07-27T21:46:41Z"}