{"id":727,"date":"2014-08-21T10:43:45","date_gmt":"2014-08-21T14:43:45","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/fgarvan\/?page_id=727"},"modified":"2026-03-19T08:11:17","modified_gmt":"2026-03-19T12:11:17","slug":"semnt","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/fgarvan\/teaching\/semnt\/","title":{"rendered":"MAT 6932 &#8211; Seminar in Number Theory"},"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\">MAT 6932 &#8211; Seminar in Number Theory<\/h1>\n\n\n\n<p>Fall 2014 Section 0978 and Spring 2015 Section 1239<\/p>\n\n\n\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"2\">\n<tbody>\n<tr>\n<td valign=\"top\" bgcolor=\"#99ccff\"><\/td>\n<td rowspan=\"2\" align=\"left\" valign=\"top\" width=\"600\">\n<table border=\"0\" width=\"95%\" cellspacing=\"0\" cellpadding=\"20\" bgcolor=\"#cccccc\">\n<tbody>\n<tr>\n<td>Current Announcements\n<ul>\n<li>Wed Jan 28: We continue with Bringmann and Ono&#8217;s paper. Later we will relate it to Zagier&#8217;s paper <a href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/content-removed\/\">Ramanujan&#8217;s mock theta functions and their applications [d&#8217;apr\u00e8s Zwegers and Bringmann-Ono]<\/a>.<\/li>\n<li>Fri Jan 16: We continue the Seminar by first working through<br>\nBringmann and Ono&#8217;s paper Dyson&#8217;s ranks and Maass forms<br>\nAnnals of Mathematics, 171 (2010), pp. 419-449<\/li>\n<li>Fri Oct 10: Proved Mittag-Leffler Theorem. Continued with the proof of Zwegers&#8217;s Lemma 3.3.<\/li>\n<li>Fri Oct 3: Showed equivalence of Watson and Zwegers version of Theorem. Proof identity at the end of Watson&#8217;s paper. Did Lemma from Atkin and Swinnerton-Dyer&#8217;s paper to prepare for proof of Watson&#8217;s quintuple product identity which is connected with Zwegers&#8217;s functions g0,g1 and g2.<\/li>\n<li>Wed Oct 1: Finished proof of Watson&#8217;s transformation formula for f(q). Started work on Zwegers&#8217; paper on Real Analytic Modular Forms.<\/li>\n<li>Mon Sep 22: We proved another identity for the 3rd order functions phi(-q) and psi(-q) again using Chan&#8217;s identity.<\/li>\n<li>Fri Sep 19: We used a case of S.H. Chan&#8217;s identity to prove an identity for the 3rd functions phi(q) and f(q).<\/li>\n<li>Fri Sep 12: We discussed some of Homework 2 problems including an overview of the Delta function as a modular form of weight 12. We showed that the proof of the even k case will follow from an identity of Zudilin. This involves Watson&#8217;s q-Whipple transformation which we started.<\/li>\n<li>Wed Sep 10: Bilateral form of the rank function. Another identity for the crank function using Bailey&#8217;s 6phi5.<\/li>\n<li>Mon Sep 8: Combinatorics of the crank function and introduction to basic hypergeometric series. Stated Bailey&#8217;s 2psi2 transformation and started proof of an identity of Ramanujan. Collected Homework 1.<\/li>\n<li>Fri Sep 5: Started work on Zudilin&#8217;s proof of the Folsom-Ono-Rhoades result for f(q). Discussed the rank and crank functions.<\/li>\n<li>Fri Aug 29: Applied Poisson Summation Formula to prove Ramanujan&#8217;s result (A) for the partition function. Homework 1 handed out. Defined the Dedekind eta-function.<\/li>\n<li>Wed Aug 27: We proved Jacobi&#8217;s Triple Product Identity and derived the Poisson Summation Formula.<\/li>\n<li>Mon Aug 25: First day of classes. <span style=\"text-decoration: underline;\"><a href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/teaching\/sumnt\/syllabus\">Syllabus<\/a><\/span> and<br>\n<span style=\"text-decoration: underline;\"><a href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/content-removed\/\">Ramanujan&#8217;s Last Letter<\/a><\/span> handed out. We discussed Ramanujan&#8217;s functions (A) and (B). We proved Euler&#8217;s Theorem for the generating function of the partition function. We introduced concept of Durfee square to explain that (B)<br>\nis the generating function of partitions into parts where difference between parts is at least 2. See Q-SERIES NOTES (below) for more detail on some of these topics.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<td align=\"left\" valign=\"top\">\n<ul>\n<li><a title=\"MAT 6932 syllabus\" href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/teaching\/semnt\/syllabus\/\" target=\"_blank\" rel=\"noopener\">SYLLABUS<\/a><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n<p>LINKS:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.youtube.com\/watch?v=OARGZ1xXCxs\">Ramanujan: Letters from an Indian Clerk [VIDEO]<\/a><\/li>\n<li><a href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/content-removed\/\">Ramanujan&#8217;s Last Letter<\/a><\/li>\n<li><a href=\"http:\/\/dx.doi.org\/10.1112\/jlms\/s1-11.1.55\">The Final Problem<\/a> (Watson 1936)<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1309.2053\">On three theorems of Folsom, Ono and Rhoades<\/a> (Zudilin, 2013)<\/li>\n<li>Ramanujan&#8217;s radial limits (Folsom, Ono, Rhoades, 2013)<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1309.4162\">Ramanujan&#8217;s radial limits and mixed mock modular bilateral $q$-hypergeometric series<\/a> (Mortenson, 2013)<\/li>\n<li><a title=\"Mock theta functions and real analytic modular forms\" href=\"http:\/\/www.researchgate.net\/publication\/228417846_Mock_-functions_and_real_analytic_modular_forms\">Mock theta functions and real analytic modular forms<\/a> (Zwegers, 2000)<\/li>\n<li><a href=\"https:\/\/people.clas.ufl.edu\/fgarvan\/content-removed\/\">Ramanujan&#8217;s mock theta functions and their applications (d&#8217;apres Zwegers and Bringmann-Ono)<\/a> (Zagier, 2007)<\/li><\/ul>\n\n\n\n<p>PERSONALITIES:<br>\n<a href=\"http:\/\/www-groups.dcs.st-and.ac.uk.\/~history\/Biographies\/Ramanujan.html\">RAMANUJAN<\/a><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">NOTES FROM MY Q-SERIES COURSE<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch1\">Chapter 1<\/a> &#8211; Partitions and Infinite Products<\/li>\n<li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch2\">Chapter 2<\/a> &#8211; Infinite Series Generating Functions and Hypergeometric Series<\/li>\n<li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch3\">Chapter 3<\/a> &#8211; Ramanujan&#8217;s Partition Congruences<\/li>\n<li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch4\">Chapter 4<\/a> &#8211; Restricted Partitions and Permutations<\/li>\n<li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch5\">Chapter 5<\/a> &#8211; Identities of the Rogers-Ramanujan Type<\/li>\n<li><a href=\"http:\/\/qseries.org\/fgarvan\/qs\/notes\/ch6\">Chapter 6<\/a> &#8211; Sums of Squares<\/li><\/ul>\n\n\n\n<p>Old News<\/p>\n\n\n\n<table border=\"0\" width=\"95%\" cellspacing=\"0\" cellpadding=\"20\" bgcolor=\"#cccccc\">\n<tbody>\n<tr>\n<td>\n<ul>\n<li>No news<\/li>\n<\/ul>\n<p>Last update made October 1, 2014, 8:45pm.<\/p><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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":140,"featured_media":0,"parent":151,"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-727","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/pages\/727","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/users\/140"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/comments?post=727"}],"version-history":[{"count":11,"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/pages\/727\/revisions"}],"predecessor-version":[{"id":3592,"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/pages\/727\/revisions\/3592"}],"up":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/pages\/151"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/fgarvan\/wp-json\/wp\/v2\/media?parent=727"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}