{"id":3307,"date":"2023-10-30T15:38:57","date_gmt":"2023-10-30T22:38:57","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=3307"},"modified":"2023-10-30T15:38:57","modified_gmt":"2023-10-30T22:38:57","slug":"slope-gap-distributions-of-translation-surfaces-taylor-mcadam-pomona-college","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/slope-gap-distributions-of-translation-surfaces-taylor-mcadam-pomona-college\/","title":{"rendered":"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College)"},"content":{"rendered":"<p><b>Title:<\/b>\u00a0Slope gap distributions of translation surfaces<\/p>\n<p><strong>Speaker:<\/strong> Taylor McAdam, Department of Mathematics, Pomona College<\/p>\n<div><b>Abstract:<\/b>\u00a0How \u201crandom\u201d are the rational numbers? To make sense of this\u00a0question, let us consider the set of Farey fractions of level n\u2014that is, the\u00a0rational\u00a0numbers between 0 and 1 with denominator at most n. It turns out that these\u00a0distribute uniformly in the unit interval as n goes to infinity, which\u00a0would suggest\u00a0they appear to be quite random. However, we may consider a finer test of\u00a0randomness by considering the distribution of\u00a0<i>gaps<\/i> between consecutive Farey\u00a0fractions as n tends to infinity. To investigate this, we will first\u00a0realize the Farey fractions as the slopes of\u00a0geodesic paths on the (square) flat\u00a0torus\u2014a geometric object obtained by gluing the opposite edges of a square\u00a0together. We will then define the\u00a0horocycle flow on the space of all flat tori,\u00a0which will allow us to study our question about gaps between Farey fractions\u00a0via a dynamical system.\u00a0Finally, we will see how this method can be generalized\u00a0to study the slope gap distributions for paths on a larger class of geometric\u00a0objects called\u00a0translation surfaces and discuss results on the collection\u00a0of surfaces obtained by gluing together opposite edges of the regular 2n-gon.<\/div>\n<div>\n<hr \/>\n<div>\n<div class=\"page\" title=\"Page 1\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Taylor McAdam graduated with a Bachelor\u2019s degree in mathematics from Harvey Mudd College in 2013 before starting a doctoral program at University of Texas at Austin. In 2017, she transferred to the University of California San Diego, where she received her PhD in mathematics in 2019 under the supervision of Amir Mohammadi. She was an NSF Postdoctoral Fellow at Yale University from 2019 to 2023, before joining the faculty at Pomona College in 2023 as a Visiting Assistant Professor. Her research interests lie at the intersection of dynamical systems, geometry, and number theory, and she is passionate about undergraduate math education and building inclusive mathematical communities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Title:\u00a0Slope gap distributions of translation surfaces Speaker: Taylor McAdam, Department of Mathematics, Pomona College Abstract:\u00a0How \u201crandom\u201d are the rational numbers? To make sense of this\u00a0question, let us consider the set [&hellip;]<\/p>\n","protected":false},"author":204,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[12],"class_list":["post-3307","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-colloquium","cat_colloquium"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"og:description\" content=\"Title:\u00a0Slope gap distributions of translation surfaces Speaker: Taylor McAdam, Department of Mathematics, Pomona College Abstract:\u00a0How \u201crandom\u201d are the rational numbers? To make sense of this\u00a0question, let us consider the set [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&amp;p=3307\" \/>\n<meta property=\"og:site_name\" content=\"Claremont Center for the Mathematical Sciences\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?post_type=tribe_events&p=3307\",\"url\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?post_type=tribe_events&p=3307\",\"name\":\"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/#website\"},\"datePublished\":\"2023-10-30T22:38:57+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?post_type=tribe_events&p=3307#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?post_type=tribe_events&p=3307\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?post_type=tribe_events&p=3307#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Events\",\"item\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/events\\\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/#website\",\"url\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/\",\"name\":\"Claremont Center for the Mathematical Sciences\",\"description\":\"Proudly Serving the Math Community at the Claremont Colleges Since 2007\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/colleges.claremont.edu\\\/ccms\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307","og_locale":"en_US","og_type":"article","og_title":"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences","og_description":"Title:\u00a0Slope gap distributions of translation surfaces Speaker: Taylor McAdam, Department of Mathematics, Pomona College Abstract:\u00a0How \u201crandom\u201d are the rational numbers? To make sense of this\u00a0question, let us consider the set [&hellip;]","og_url":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307","og_site_name":"Claremont Center for the Mathematical Sciences","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307","url":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307","name":"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College) - Claremont Center for the Mathematical Sciences","isPartOf":{"@id":"https:\/\/colleges.claremont.edu\/ccms\/#website"},"datePublished":"2023-10-30T22:38:57+00:00","breadcrumb":{"@id":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3307#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/colleges.claremont.edu\/ccms\/"},{"@type":"ListItem","position":2,"name":"Events","item":"https:\/\/colleges.claremont.edu\/ccms\/events\/"},{"@type":"ListItem","position":3,"name":"Slope Gap Distributions of Translation Surfaces (Taylor McAdam, Pomona College)"}]},{"@type":"WebSite","@id":"https:\/\/colleges.claremont.edu\/ccms\/#website","url":"https:\/\/colleges.claremont.edu\/ccms\/","name":"Claremont Center for the Mathematical Sciences","description":"Proudly Serving the Math Community at the Claremont Colleges Since 2007","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/colleges.claremont.edu\/ccms\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"ticketed":false,"_links":{"self":[{"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/tribe_events\/3307","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/tribe_events"}],"about":[{"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/types\/tribe_events"}],"author":[{"embeddable":true,"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/users\/204"}],"version-history":[{"count":0,"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/tribe_events\/3307\/revisions"}],"wp:attachment":[{"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/media?parent=3307"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/tags?post=3307"},{"taxonomy":"tribe_events_cat","embeddable":true,"href":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/wp\/v2\/tribe_events_cat?post=3307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}