{"id":2940,"date":"2022-07-05T06:54:20","date_gmt":"2022-07-04T21:54:20","guid":{"rendered":"https:\/\/www.ibs.re.kr\/ecopro\/?page_id=2940"},"modified":"2022-10-10T11:40:38","modified_gmt":"2022-10-10T02:40:38","slug":"summer-2022","status":"publish","type":"page","link":"https:\/\/www.ibs.re.kr\/ecopro\/summer-2022\/","title":{"rendered":"Summer 2022"},"content":{"rendered":"\n<ul class=\"wp-block-list\"><li>Time: August 1 &#8211; 6, Korean time:<br>I. Expanders: 9:30am-10:50am;<br>II. Topological combinatorics: 11:00am-12:20pm;<br>III. Convexity spaces: 2:30pm-3:50pm. <\/li><li><strong><span style=\"color: #0693e3;\" class=\"stk-highlight\">Zoom: 819 0982 4573, PW: 202208<\/span><\/strong><\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><span style=\"color: #222222;background-color: #ff6900\" class=\"stk-highlight\">Announcement<\/span><\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li><strong><span style=\"color: #cf2e2e;\" class=\"stk-highlight\">Time change: on August 4th (Thursday), Convexity spaces will be at 11am, Topological combinatorics will be at 2:30pm.<\/span><\/strong><\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Sparsifiers and expanders<\/h2>\n\n\n<div style=\"gap: 20px;\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-8b2c3def-e1ca-4751-9b35-fde838900e47\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Expander2-JaehoonKim.pdf\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #cf2e2e; --ub-button-border: 3px solid #cf2e2e; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 100px; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Note<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV1jN4y1j7iP?share_source=copy_web\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #0693e3; --ub-button-border: 3px solid #0693e3; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 20%; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Videos<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<ul class=\"wp-block-list\"><li>8\/1: Why Laplacian is useful: applications in image processing, analogue of Laplacian operator on Euclidean space.<\/li><li>8\/2: Random walks on non-regular graphs, spectral sparsifiers with $O(n\\log n)$ edges.<\/li><li>8\/3: Spectral sparsifiers with $O(n\\log n)$ edges, Interlacing polynomials.<\/li><li>8\/4: Spectral sparsifiers with linear many edges.<\/li><li>8\/5: Existence of bipartite Ramanujan graphs using 2-covers.<\/li><li>8\/6: Existence of bipartite Ramanujan graphs by taking union of perfect matchings.<\/li><\/ul>\n\n\n\n<div class=\"wp-block-stackable-image stk-block-image stk-block stk-b356707\" data-block-id=\"b356707\"><figure class=\"stk-img-wrapper stk-image--shape-stretch\"><img loading=\"lazy\" decoding=\"async\" class=\"stk-img wp-image-3089\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing.jpg\" width=\"1057\" height=\"468\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing.jpg 1057w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing-300x133.jpg 300w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing-1024x453.jpg 1024w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing-768x340.jpg 768w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/interlacing-600x266.jpg 600w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Topological combinatorics<\/h2>\n\n\n<div style=\"gap: 20px;\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-043e3d99-2732-40af-b1f1-821daca5eca8\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.overleaf.com\/read\/yxxqpsncpzdk\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #cf2e2e; --ub-button-border: 3px solid #cf2e2e; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 100px; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Note<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV1mY4y1P7bV?share_source=copy_web\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #0693e3; --ub-button-border: 3px solid #0693e3; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 20%; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Videos<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<ul class=\"wp-block-list\"><li>8\/1: Homotopy, Homology, Nerve theorem, good cover, Topological Helly.  <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/lecturenote1_jinhakim.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/2: Three basic tools in topological combinatorics and their equivalence: Brouwer Fixed-Point Theorem\/Sperner&#8217;s Lemma\/KKM Theorem.  <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/lecturenote2_jinhakim.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/3: Brouwer implies weak Sperner, and applications of KKM theorem:(p,q)-theorem\/Line&nbsp;transversal\/Colorful KKM theorem. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/lecturenote3_jinhakim.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/4: Hall\u2019s theorem for hypergraphs, proof based on Sperner\u2019s lemma to an economically hierarchic triangulation of a simplex. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Topological-Combinatorics-4.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/5: Ryser&#8217;s conjecture for tripartite 3-uniform hypergraphs, homology nerve theorem, topological Hall theorem. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/lecturenote2_minkikim.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/6: Homology nerve theorem, Topological Hall theorem (correction), $d$-collapsible\/Leray complexes, Topological colorful Helly theorem, Alexander duality theorem. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/lecturenote3_minkikim.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><\/ul>\n\n\n\n<div class=\"wp-block-jetpack-tiled-gallery aligncenter is-style-rectangular\"><div class=\"tiled-gallery__gallery\"><div class=\"tiled-gallery__row\"><div class=\"tiled-gallery__col\" style=\"flex-basis:36.14342%\"><figure class=\"tiled-gallery__item\"><img decoding=\"async\" srcset=\"https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb2022.jpg?strip=info&#038;w=447&#038;ssl=1 447w\" alt=\"\" data-height=\"421\" data-id=\"3090\" data-link=\"https:\/\/www.ibs.re.kr\/ecopro\/top-comb2022\/\" data-url=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb2022.jpg\" data-width=\"447\" src=\"https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb2022.jpg?ssl=1\" data-amp-layout=\"responsive\"\/><\/figure><\/div><div class=\"tiled-gallery__col\" style=\"flex-basis:63.85658%\"><figure class=\"tiled-gallery__item\"><img decoding=\"async\" srcset=\"https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=600&#038;ssl=1 600w,https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=900&#038;ssl=1 900w,https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=1200&#038;ssl=1 1200w,https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=1500&#038;ssl=1 1500w,https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=1800&#038;ssl=1 1800w,https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?strip=info&#038;w=1832&#038;ssl=1 1832w\" alt=\"\" data-height=\"974\" data-id=\"3085\" data-link=\"https:\/\/www.ibs.re.kr\/ecopro\/top-comb\/\" data-url=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png\" data-width=\"1832\" src=\"https:\/\/i0.wp.com\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/top-comb-1024x544.png?ssl=1\" data-amp-layout=\"responsive\"\/><\/figure><\/div><\/div><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Convexity spaces<\/h2>\n\n\n<div style=\"gap: 20px;\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-e1e82d84-126a-41a9-95cf-a778730906ab\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.overleaf.com\/read\/kxqmwknfcxkb\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #cf2e2e; --ub-button-border: 3px solid #cf2e2e; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 100px; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Note<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV1AG4y1v7iK?share_source=copy_web\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #0693e3; --ub-button-border: 3px solid #0693e3; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 20%; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Videos<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<ul class=\"wp-block-list\"><li>8\/1: Examples of convexity spaces: convex lattices sets, subgroups, subtrees, subcubes, invariants abstracting basic properties of Euclidean convexity space: Radon\/Carath\u00e9odory\/Helly. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp1.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/2: Radon number of subcubes in $\\{0,1\\}^n$ is about $\\log_2 n$, weak $\\varepsilon$-net. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp2.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/3: weak $\\varepsilon$-net \u21d2 Radon, colorful\/fractional Helly, Tverberg number. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp3.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/4: Helly + Tverberg \u21d2 colorful Helly \u21d2 fractional Helly, Radon \u21d2 fractional Helly, selection \u21d2 weak $\\varepsilon$-net. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp4.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/5: Fractional Helly \u21d2 Tverberg using hypergraph matching, fractional Helly + Tverberg \u21d2 selection, equivalent formulation of weak $\\varepsilon$-net: bounding transversal number via fractional transversal number. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp5.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><li>8\/6: Radon \u21d2 weak $\\varepsilon$-net in separable convexity space via VC dim. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/Conv-sp6.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">Tablet note<\/span><\/a><\/strong><\/li><\/ul>\n\n\n\n<div class=\"wp-block-stackable-image stk-block-image stk-block stk-cf7c18e\" data-block-id=\"cf7c18e\"><figure class=\"stk-img-wrapper stk-image--shape-stretch\"><img loading=\"lazy\" decoding=\"async\" class=\"stk-img wp-image-3088\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/conv-sp2022.jpg\" width=\"890\" height=\"583\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/conv-sp2022.jpg 890w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/conv-sp2022-300x197.jpg 300w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/conv-sp2022-768x503.jpg 768w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/08\/conv-sp2022-600x393.jpg 600w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/figure><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-stackable-image stk-block-image stk-block stk-49f8137\" data-block-id=\"49f8137\"><figure class=\"stk-img-wrapper stk-image--shape-stretch\"><img loading=\"lazy\" decoding=\"async\" class=\"stk-img wp-image-2986\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022.png\" width=\"2480\" height=\"3508\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022.png 2480w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-212x300.png 212w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-724x1024.png 724w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-768x1086.png 768w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-1086x1536.png 1086w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-1448x2048.png 1448w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022-283x400.png 283w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Time: August 1 &#8211; 6, Korean time:I. Expanders: 9:30am-10:50am;II. Topological combinatorics: 11:00am-12:20pm;III. Convexity spaces: 2:30pm-3:50pm. Zoom: 819 0982 4573, PW: 202208 Announcement Time change: on August 4th (Thursday), Convexity spaces &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/summer-2022\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Summer 2022&#8221;<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":2986,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","_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,"footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"class_list":["post-2940","page","type-page","status-publish","has-post-thumbnail","hentry"],"featured_image_src":"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2022\/07\/summer-school-2022.png","yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Summer 2022 - Extremal Combinatorics and Probability Group<\/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:\/\/www.ibs.re.kr\/ecopro\/summer-2022\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Summer 2022 - Extremal Combinatorics and Probability Group\" \/>\n<meta property=\"og:description\" content=\"Time: August 1 &#8211; 6, Korean time:I. Expanders: 9:30am-10:50am;II. Topological combinatorics: 11:00am-12:20pm;III. Convexity spaces: 2:30pm-3:50pm. 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