{"id":2580,"date":"2023-03-15T12:32:04","date_gmt":"2023-03-15T10:32:04","guid":{"rendered":"https:\/\/mathematics.haifa.ac.il\/?p=2580"},"modified":"2023-03-22T11:17:05","modified_gmt":"2023-03-22T09:17:05","slug":"colloquium-tuesday-january-21-2023-speaker-christopher-phillips-oregon-title-relations-between-dynamics-and-c-algebras-mean-dimension-and-radius-of-comparison","status":"publish","type":"post","link":"https:\/\/mathematics.haifa.ac.il\/?p=2580","title":{"rendered":"Colloquium: Tuesday March 21, 2023. Speaker: Christopher\u00a0Phillips (Oregon). Title: &#8220;Relations between dynamics and C*-algebras: Mean dimension and radius of comparison&#8221;."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">room 614, Science &amp; Education building.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A zoom link for our meetings is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/us02web.zoom.us\/j\/83337601824\">https:\/\/us02web.zoom.us\/j\/83337601824<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Speaker :&nbsp;<strong>Christopher&nbsp;Phillips (Oregon)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Date :&nbsp;<strong>Tuesday, 21st of March, 2023.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Time :&nbsp;<strong>14:00<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Title:<\/strong>&nbsp;Relations between dynamics and C*-algebras: Mean dimension and<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">radius of comparison.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Abstract:&nbsp;<\/strong>For an action of an amenable group <img loading=\"lazy\" decoding=\"async\" width=\"11\" height=\"11\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=G\" alt=\"G\">on a compact metric space~<img loading=\"lazy\" decoding=\"async\" width=\"13\" height=\"11\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=X\" alt=\"X\">, the mean dimension <img loading=\"lazy\" decoding=\"async\" width=\"86\" height=\"16\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7b%5Cmathrm%7bmdim%7d%7d%09(G,%09X)\" alt=\"{\\mathrm{mdim}} (G, X)\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">was introduced by Lindenstrauss and Weiss.<br>It is designed so that the mean dimension of the shift on<br><img loading=\"lazy\" decoding=\"async\" width=\"59\" height=\"17\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=(%5b0,%091%5d%5Ed)%5EG\" alt=\"([0, 1]^d)^G\">is~<img loading=\"lazy\" decoding=\"async\" width=\"8\" height=\"11\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=d\" alt=\"d\">.&nbsp;Its motivation was unrelated to C*-algebras.<br><br>The radius of comparison <img loading=\"lazy\" decoding=\"async\" width=\"36\" height=\"16\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7b%5Cmathrm%7brc%7d%7d%09(A)\" alt=\"{\\mathrm{rc}} (A)\">of a C*-algebra~<img loading=\"lazy\" decoding=\"async\" width=\"11\" height=\"11\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=A\" alt=\"A\"><br>was introduced by Toms to distinguish counterexamples<br>in the Elliott classification program.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The algebras he used have nothing to do with dynamics.<br><br>A construction called the crossed product associates<br>a C*-algebra <img loading=\"lazy\" decoding=\"async\" width=\"64\" height=\"16\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=C%5E*%09(G,%09X)\" alt=\"C^* (G, X)\">to a dynamical system <img loading=\"lazy\" decoding=\"async\" width=\"44\" height=\"16\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=(G,%09X)\" alt=\"(G, X)\">.<br>Despite the apparent lack of connection between these concepts,<br>there is significant evidence for the conjecture that<br><img loading=\"lazy\" decoding=\"async\" width=\"210\" height=\"19\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7b%5Cmathrm%7brc%7d%7d%09(%09C%5E*%09(G,%09X)%09)%09=%09%5Cfrac%7b1%7d%7b2%7d%09%7b%5Cmathrm%7bmdim%7d%7d%09(G,%09X)\" alt=\"{\\mathrm{rc}} ( C^* (G, X) ) = \\frac{1}{2} {\\mathrm{mdim}} (G, X)\"><br>when the action is free and minimal.<br>We will explain the concepts above; no previous knowledge<br>of mean dimension, C*-algebras, or radius of comparison will be assumed.<br>Then we describe some of the evidence.<br>In particular, we give the first general partial results<br>towards the direction<br><img loading=\"lazy\" decoding=\"async\" width=\"210\" height=\"19\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7b%5Cmathrm%7brc%7d%7d%09(%09C%5E*%09(G,%09X)%09)%09%5Cgeq%09%5Cfrac%7b1%7d%7b2%7d%09%7b%5Cmathrm%7bmdim%7d%7d%09(G,%09X)\" alt=\"{\\mathrm{rc}} ( C^* (G, X) ) \\geq \\frac{1}{2} {\\mathrm{mdim}} (G, X)\">.<br>We don&#8217;t get the exact conjectured bound,<br>but we get nontrivial results for many of the known examples<br>of free minimal systems with <img loading=\"lazy\" decoding=\"async\" width=\"117\" height=\"16\" src=\"https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=%7b%5Cmathrm%7bmdim%7d%7d%09(G,%09X)%09%3e%090\" alt=\"{\\mathrm{mdim}} (G, X) > 0&#8243;>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is joint work with Ilan Hirshberg.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>room 614, Science &amp; Education building. A zoom link for our meetings is: https:\/\/us02web.zoom.us\/j\/83337601824 Speaker :&nbsp;Christopher&nbsp;Phillips (Oregon) Date :&nbsp;Tuesday, 21st of March, 2023. Time :&nbsp;14:00 Title:&nbsp;Relations between dynamics and C*-algebras: Mean dimension and radius of comparison. Abstract:&nbsp;For an action of&#8230;<br \/><a class=\"read-more-button\" href=\"https:\/\/mathematics.haifa.ac.il\/?p=2580\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-2580","post","type-post","status-publish","format-standard","hentry","category-colloquium"],"_links":{"self":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2580","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2580"}],"version-history":[{"count":2,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2580\/revisions"}],"predecessor-version":[{"id":2582,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2580\/revisions\/2582"}],"wp:attachment":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2580"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}