{"id":3893,"date":"2023-04-27T10:08:24","date_gmt":"2023-04-27T01:08:24","guid":{"rendered":"https:\/\/www.ibs.re.kr\/ecopro\/?page_id=3893"},"modified":"2023-07-28T10:33:25","modified_gmt":"2023-07-28T01:33:25","slug":"summer-2023","status":"publish","type":"page","link":"https:\/\/www.ibs.re.kr\/ecopro\/summer-2023\/","title":{"rendered":"Summer 2023"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li>Time: 9:00-10:30am (Beijing), 10-11:30am(Seoul)<\/li>\n\n\n\n<li><strong>July 17-21<\/strong>, The method of hypergraph containers.<br><em>@ <span style=\"color: #8e1e1e;\" class=\"stk-highlight\">IBS Science Culture Center, <strong>S221<\/strong> (2nd floor), Daejeon, South Korea.<\/span><\/em><\/li>\n\n\n\n<li><strong>July 24-30<\/strong>, Graph limits and flag algebras.<br><em>@<\/em> <em><span style=\"color: #8e1e1e;\" class=\"stk-highlight\">Dongmingzhu Building 112, Shandong University, Jinan, China.<\/span><\/em><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><span style=\"color: #0693e3;\" class=\"stk-highlight\">Zoom: 346 934 4087, PW: 202307<\/span><\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">The method of hypergraph containers<\/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-630741f1-773d-4deb-ba92-0e7230d21f75\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/Prague-school.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\/BV1614y197yF\/\" 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 class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/container-exercises.pdf\" target=\"_blank\" rel=\"noopener noreferrer\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #313131; --ub-button-border: 3px solid #313131; --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: 10px; \">\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>Exercises<\/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\">\n<li>7\/17 Lecture 1: Classical results about triangle-free graphs \u2013 Mantel\u2019s and Ramsey\u2019s theorems (without proofs); the binomial random graph $G_{n,p}$; Mantel\u2019s and Ramsey\u2019s theorems relative to a (random) graph; the Erd\u0151s\u2013Kleitman\u2013Rothschild theorem on the number of triangle-free graphs and the structure of a random triangle-free graph (without proof); Hoeffding\u2019s theorem (aka Chernoff\u2019s bound); maximal triangle-free graphs; a weak container theorem for triangle-free graphs; the supersaturation theorem of Erd\u0151s and Simonovits (without proof); derivation of an upper bound on the number of triangle-free graphs from the weak container theorem for triangle-free graphs and the supersaturation theorem<\/li>\n\n\n\n<li>7\/18 Lecture 2: Proof of the Erd\u0151s\u2013Simonovits supersaturation theorem; proof of the random analogue of Mantel\u2019s theorem (for $p \\gg n^{-1\/2} \\log n$); Ramsey multiplicity; proof of the random analogue of Ramsey\u2019s theorem for triangles (for $p \\gg n^{-1\/2} \\log n$); necessary conditions on the edge probability $p$ for random analogues of Mantel\u2019s theorem and Ramsey\u2019s theorem for triangles <\/li>\n\n\n\n<li>7\/19 Lecture 3: Container theorem for triangle-free graphs; proof of Mantel\u2019s theorem relative to $G_{n,p}$ under the optimal assumption $p \\gg n^{-1\/2}$; Simonovits\u2019s stability theorem for triangle-free graphs (F\u00fcredi\u2019s proof); the triangle-removal lemma of Ruzsa and Szemer\u00e9di (without proof); stability theorem for almost-triangle-free graphs; proof of Simonovits\u2019s stability theorem relative to $G_{n,p}$<\/li>\n\n\n\n<li>7\/20 Lecture 4: Independent sets in hypergraphs; the general hypergraph container theorem; derivation of the container theorem for triangle-free graphs; optimality of assumptions in the hypergraph container theorem on the example of the general Tur\u00e1n problem in random graphs; the existence of induced Ramsey graphs via the hypergraph container theorem<\/li>\n\n\n\n<li>7\/21 Lecture 5: The $(4,3)$-problem \u2013 subsets of points in general position in point sets without collinear $4$-tuples; a randomised construction of such n-element sets whose largest subsets in general position has no more than $n^{5\/6+o(1)}$ points (the Balogh\u2013Solymosi theorem)<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Graph limits and flag algebras<\/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-c35626ef-52df-4044-8e09-bfa548ac80f9\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV13c411w7tT\/\" 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\">\n<li>7\/24 Lecture 1: convergence of graph sequences, limits of dense graphs, graphons, cut distance <br><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/lecture-24.07.pdf\"><span style=\"color: #c81313;\" class=\"stk-highlight\">Note<\/span><\/a><\/strong>  <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/exercises1.pdf\"><span style=\"color: #1c25d2;\" class=\"stk-highlight\">Exercise<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/25 Lecture 2: definition of flag algebras<br><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/lecture-25.07.pdf\"><span style=\"color: #c81313;\" class=\"stk-highlight\">Note<\/span><\/a><\/strong>  <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/exercises2.pdf\"><span style=\"color: #1c25d2;\" class=\"stk-highlight\">Exercise<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/26 Lecture 3: proof of Mathel&#8217;s theorem using flag algebras, setting a semidefinite programming for automatic flag algebras proofs<br><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/lecture-26.07.pdf\"><span style=\"color: #c81313;\" class=\"stk-highlight\">Note<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/27 Lecture 4: rounding the numerical solution, using Flagmatic for computations in flag algebras, example usage of flag algebras for oriented graphs<br><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/lecture-27.07.pdf\"><span style=\"color: #c81313;\" class=\"stk-highlight\">Note<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/25 Lecture 5: applications of flag algebras, differential method<br><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/07\/lecture-28.07.pdf\"><span style=\"color: #c81313;\" class=\"stk-highlight\">Note<\/span><\/a><\/strong><\/li>\n<\/ul>\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-3929\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/05\/summer2023-poster-scaled.jpeg\" width=\"1810\" height=\"2560\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/05\/summer2023-poster-scaled.jpeg 1810w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/05\/summer2023-poster-212x300.jpeg 212w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2023\/05\/summer2023-poster-724x1024.jpeg 724w, 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