{"id":13451,"date":"2020-06-17T17:39:04","date_gmt":"2020-06-18T00:39:04","guid":{"rendered":"http:\/\/www.preposterousuniverse.com\/blog\/?p=13451"},"modified":"2020-06-21T09:11:00","modified_gmt":"2020-06-21T16:11:00","slug":"the-biggest-ideas-in-the-universe-13-geometry-and-topology","status":"publish","type":"post","link":"https:\/\/preposterousuniverse.com\/blog\/2020\/06\/17\/the-biggest-ideas-in-the-universe-13-geometry-and-topology\/","title":{"rendered":"The Biggest Ideas in the Universe |13. Geometry and Topology"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Cheating by having two ideas this week. But they are big ones. We talk about how &#8220;parallel transport&#8221; of vectors allows us to define curvature intrinsically, and leads us to the Riemann curvature tensor. Then we move to topology, in particular homotopy groups, which show up in physics all the time.<\/p>\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"lyte-wrapper\" title=\"The Biggest Ideas in the Universe | 13. Geometry and Topology\" style=\"width:640px;max-width:100%;margin:5px;\"><div class=\"lyMe\" id=\"WYL_kp1k90zNVLc\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><div><meta itemprop=\"thumbnailUrl\" content=\"https:\/\/i.ytimg.com\/vi\/kp1k90zNVLc\/hqdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/kp1k90zNVLc\" \/><meta itemprop=\"duration\" content=\"PT1H26M9S\" \/><meta itemprop=\"uploadDate\" content=\"2020-06-16T14:27:50Z\" \/><\/div><div id=\"lyte_kp1k90zNVLc\" data-src=\"https:\/\/i.ytimg.com\/vi\/kp1k90zNVLc\/hqdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">The Biggest Ideas in the Universe | 13. Geometry and Topology<\/div><\/div><div class=\"play\"><\/div><div class=\"ctrl\"><div class=\"Lctrl\"><\/div><div class=\"Rctrl\"><\/div><\/div><\/div><noscript><a href=\"https:\/\/youtu.be\/kp1k90zNVLc\" rel=\"nofollow\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i.ytimg.com\/vi\/kp1k90zNVLc\/0.jpg\" alt=\"The Biggest Ideas in the Universe | 13. Geometry and Topology\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is Idea #13, &quot;Geometry and Topology.&quot; Yes that&#039;s two ideas, and furthermore they&#039;re from math more than from science, but we&#039;ll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called &quot;homotopy groups.&quot; My web page: http:\/\/www.preposterousuniverse.com\/ My YouTube channel: https:\/\/www.youtube.com\/c\/seancarroll Mindscape podcast: http:\/\/www.preposterousuniverse.com\/podcast The Biggest Ideas playlist: https:\/\/www.youtube.com\/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX Blog posts for the series: http:\/\/www.preposterousuniverse.com\/blog\/category\/biggest-ideas-in-the-universe\/ Background image by RyoThorn at DeviantArt: https:\/\/www.deviantart.com\/ryothorn\/art\/Steamworks-to-my-own-oblivion-147929840 #science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px;\"><\/div><figcaption><\/figcaption><\/figure>\n\n\n<p class=\"wp-block-paragraph\">And here is the associated Q&amp;A video, where we talk a bit about embeddings, tensor components, topological defects, and more.<\/p>\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"lyte-wrapper\" title=\"The Biggest Ideas in the Universe | Q&amp;amp;A 13 - Geometry and Topology\" style=\"width:640px;max-width:100%;margin:5px;\"><div class=\"lyMe\" id=\"WYL_TXgezspaGi8\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><div><meta itemprop=\"thumbnailUrl\" content=\"https:\/\/i.ytimg.com\/vi\/TXgezspaGi8\/hqdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/TXgezspaGi8\" \/><meta itemprop=\"duration\" content=\"PT1H1M7S\" \/><meta itemprop=\"uploadDate\" content=\"2020-06-21T16:06:49Z\" \/><\/div><div id=\"lyte_TXgezspaGi8\" data-src=\"https:\/\/i.ytimg.com\/vi\/TXgezspaGi8\/hqdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">The Biggest Ideas in the Universe | Q&amp;A 13 - Geometry and Topology<\/div><\/div><div class=\"play\"><\/div><div class=\"ctrl\"><div class=\"Lctrl\"><\/div><div class=\"Rctrl\"><\/div><\/div><\/div><noscript><a href=\"https:\/\/youtu.be\/TXgezspaGi8\" rel=\"nofollow\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i.ytimg.com\/vi\/TXgezspaGi8\/0.jpg\" alt=\"The Biggest Ideas in the Universe | Q&amp;amp;A 13 - Geometry and Topology\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is the Q&amp;A video for Idea #13, &quot;Geometry and Topology.&quot; We use the excuse to dig into some details of embeddings, how to express the Riemann tensor, what maps are involved in specifying homotopy groups, plus a bit about topological defects. My web page: http:\/\/www.preposterousuniverse.com\/ My YouTube channel: https:\/\/www.youtube.com\/c\/seancarroll Mindscape podcast: http:\/\/www.preposterousuniverse.com\/podcast The Biggest Ideas playlist: https:\/\/www.youtube.com\/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX Blog posts for the series: http:\/\/www.preposterousuniverse.com\/blog\/category\/biggest-ideas-in-the-universe\/ Background image by RyoThorn at DeviantArt: https:\/\/www.deviantart.com\/ryothorn\/art\/Steamworks-to-my-own-oblivion-147929840 #science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px;\"><\/div><figcaption><\/figcaption><\/figure>","protected":false},"excerpt":{"rendered":"<p>Cheating by having two ideas this week. But they are big ones. We talk about how &#8220;parallel transport&#8221; of vectors allows us to define curvature intrinsically, and leads us to the Riemann curvature tensor. Then we move to topology, in particular homotopy groups, which show up in physics all the time. And here is the [&hellip;]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[137,28],"tags":[],"class_list":["post-13451","post","type-post","status-publish","format-standard","hentry","category-biggest-ideas-in-the-universe","category-science"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13451","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/comments?post=13451"}],"version-history":[{"count":2,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13451\/revisions"}],"predecessor-version":[{"id":13453,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13451\/revisions\/13453"}],"wp:attachment":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/media?parent=13451"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/categories?post=13451"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/tags?post=13451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}