{"id":2763,"date":"2025-10-13T00:15:33","date_gmt":"2025-10-13T00:15:33","guid":{"rendered":"https:\/\/yurilima-math.com\/?page_id=2763"},"modified":"2025-10-14T18:56:10","modified_gmt":"2025-10-14T18:56:10","slug":"lecture-notes-2","status":"publish","type":"page","link":"https:\/\/yurilima-math.com\/?page_id=2763","title":{"rendered":"Lecture Notes"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"2763\" class=\"elementor elementor-2763\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2402091b elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"2402091b\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-35ff605c\" data-id=\"35ff605c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-46c38faa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"46c38faa\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-3d4b90c4\" data-id=\"3d4b90c4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ec11118 elementor-widget elementor-widget-image\" data-id=\"ec11118\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/yurilima-math.com\/wp-content\/uploads\/elementor\/thumbs\/lecture-pxuq2aaczhbmg86nnr4ycheouhu589eup0w9bg37i0.png\" title=\"lecture\" alt=\"lecture\" loading=\"lazy\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-45b771d elementor-widget elementor-widget-spacer\" data-id=\"45b771d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-815f0a2 elementor-widget elementor-widget-heading\" data-id=\"815f0a2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Lecture Notes<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-49f4f10 elementor-widget elementor-widget-spacer\" data-id=\"49f4f10\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-85717c3 elementor-widget elementor-widget-accordion\" data-id=\"85717c3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1391\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1391\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-right\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><i class=\"fas fa-plus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><i class=\"fas fa-minus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Introdu\u00e7\u00e3o aos Sistemas Din\u00e2micos<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1391\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1391\"><p>In Portuguese, these notes compile the topics covered in the course &#8220;Introduction to Dynamical Systems&#8221; that has be taught at UFC in the last years.<\/p><p><a href=\"https:\/\/yurilima-math.com\/wp-content\/uploads\/2023\/02\/intro-SD.pdf\" target=\"_blank\" rel=\"noopener\">Lecture Notes<\/a><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1392\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1392\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-right\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><i class=\"fas fa-plus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><i class=\"fas fa-minus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Lectures on Ratner Theory<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1392\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1392\"><p>The purpose of these lecture notes was to help the author in the organization of the mini-course \u201cIntroduction to Ratner theory\u201d, held at IMPA on August 2013, as part of the activities of the thematic semester \u201cDynamics Beyond Uniform Hyperbolicity\u201d.<\/p><p>The aim of the mini-course was to introduce Ph.D. students to some beautiful topics of dynamics on homogeneous spaces, with emphasis on Ratner\u2019s theorems on the classification of orbit closures, invariant measures and equidistribution of actions of subgroups generated by unipotent elements on homogeneous spaces.<\/p><p><a href=\"https:\/\/yurilima-math.com\/wp-content\/uploads\/2022\/10\/root_ratner.pdf\" target=\"_blank\" rel=\"noopener\">Lecture Notes<\/a><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1393\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1393\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-right\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><i class=\"fas fa-plus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><i class=\"fas fa-minus\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Szemer\u00e9di's Regularity Lemma<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1393\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1393\"><p class=\"p1\">The Szemer\u00e9d regularity lemma is an important tool in discrete mathematics, specially in graph theory and additive combinatorics. The lemma can be used to theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. One example is the graph removal lemma, which can be used to prove Szemer\u00e9di&#8217;s theorem on the existence of arithmetic progressions.<\/p><p><a href=\"https:\/\/yurilima-math.com\/wp-content\/uploads\/2022\/10\/szemeredi_regularity_lemma.pdf\" target=\"_blank\" rel=\"noopener\">Lecture Notes in English<\/a><\/p><p><a href=\"https:\/\/yurilima-math.com\/wp-content\/uploads\/2022\/10\/lema_de_regularidade_de_szemere\u0301di.pdf\" target=\"_blank\" rel=\"noopener\">Lecture Notes in Portuguese<\/a><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aa95984 elementor-widget elementor-widget-spacer\" data-id=\"aa95984\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div 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class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Todos os direitos reservados @ Yuri Lima 2025<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Lecture Notes Introdu\u00e7\u00e3o aos Sistemas Din\u00e2micos In Portuguese, these notes compile the topics covered in the course &#8220;Introduction to Dynamical Systems&#8221; that has be taught at UFC in the last years. Lecture Notes Lectures on Ratner Theory The purpose of these lecture notes was to help the author in the organization of the mini-course \u201cIntroduction [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1263,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-2763","page","type-page","status-publish","has-post-thumbnail","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/pages\/2763","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yurilima-math.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2763"}],"version-history":[{"count":3,"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/pages\/2763\/revisions"}],"predecessor-version":[{"id":2789,"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/pages\/2763\/revisions\/2789"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/yurilima-math.com\/index.php?rest_route=\/wp\/v2\/media\/1263"}],"wp:attachment":[{"href":"https:\/\/yurilima-math.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}