{"id":3527,"date":"2023-01-17T12:52:32","date_gmt":"2023-01-17T17:52:32","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/boyland\/?page_id=3527"},"modified":"2026-03-19T08:10:24","modified_gmt":"2026-03-19T12:10:24","slug":"richard-hain","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/boyland\/richard-hain\/","title":{"rendered":"Richard Hain"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h1 class=\"wp-block-heading\">Richard Hain<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Hecke operators and Loops<\/h3>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">Abstract: Hecke operators were introduced by Erich Hecke in the 1930s.<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">They can be thought of as multivalued functions on the space of<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">lattices in the plane. They consequently act on certain functions on<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">this space &#8212; modular forms. In this talk I will give an accessible<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">introduction to Hecke operators and explain how they also act on the<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">free abelian group generated by the conjugacy classes in the modular<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">group SL_2(Z). Surprisingly, this leads to a non-commutative<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">generalization of the Hecke algebra. If there is sufficient time, I<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">will discuss how the Hecke action on loops leads to a Hecke action on<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">those &#8220;iterated Shimura integrals&#8221; that are constant on conjugacy<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">classes of SL_2(Z) and explain the hope that this Hecke action will<\/p>\n\n\n\n<br class=\"\">\n\n\n\n<p class=\"wp-block-paragraph\">aid in the computation of their periods.<\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":146,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-3527","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3527","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/users\/146"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/comments?post=3527"}],"version-history":[{"count":2,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3527\/revisions"}],"predecessor-version":[{"id":3816,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3527\/revisions\/3816"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/media?parent=3527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}