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SUMMARY:N. Christopher Phillips (University of Oregon)
DTSTART:20240507T190000Z
DTEND:20240507T200000Z
DTSTAMP:20260420T052852Z
UID:NYC-NCG/143
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/143/
 ">Minicourse:  An invitation to mean dimension of a dynamical system   and
  the radius of comparison of its crossed product\, I</a>\nby N. Christophe
 r Phillips (University of Oregon) as part of Noncommutative geometry in NY
 C\n\n\nAbstract\nPrerequisites (optional): \n\n1. https://sju.webex.com/re
 cordingservice/sites/sju/recording/e0819482c399103cbf7c005056812d4c/playba
 ck\n\n2. https://sju.webex.com/recordingservice/sites/sju/recording/480a92
 c95598103cae68005056819173/playback\n\n\nThe purpose of this minicourse is
  to explain the background\n(including the terms below) and some progress 
 towards the following conjecture\, relating topological dynamics to the st
 ructure of the crossed product $C^*$-algebra.\n\nLet $G$ be a countable am
 enable group\, let $X$ be a compact metrizable space\,\nand let $T$ be an 
 action of $G$ on $X$. The mean dimension $mdim ~(T)$ is a \npurely dynamic
 al invariant\, designed so that the mean dimension of the shift \non $([0\
 , 1]^d)^G$ is equal to $d$. The radius of comparison $rc ~(A)$ of a \nunit
 al $C^*$-algebra $A$ is a numerical measure of failure of comparison\nin t
 he Cuntz semigroup of $A$\, a generalization of unstable K-theory.\nIt was
  introduced to distinguish $C^*$-algebras having no connection\nwith dynam
 ics. The conjecture asserts that if $T$ is free and minimal\,\nthen $rc ~(
 C^* (G\, X\, T)) = \\frac{1}{2} ~mdim ~(T)$. The inequality\n$rc ~(C^* (G\
 , X\, T)) \\leq \\frac{1}{2} ~mdim ~(T)$ is known for \n$G = {\\mathbb{Z}}
 ^n$\, and progress towards the inequality\n$rc ~(C^* (G\, X\, T)) \\geq \\
 frac{1}{2} ~mdim ~(T)$ has been made for the known \nclasses of examples o
 f free minimal actions with nonzero mean dimension\,\nfor any countable am
 enable group $G$. The emphasis will be on the inequality\n$rc ~(C^* (G\, X
 \, T)) \\geq \\frac{1}{2} ~mdim ~(T)$\;\nthe results there are joint work 
 with Ilan Hirshberg.\n\n\nLecture 1. \n\nThis lecture will be mainly about
  dynamical systems.\nAfter an introduction\, we will review the crossed pr
 oduct $C^*$-algebra associated to a\ndynamical system\, and then describe 
 the mean dimension of a dynamical system.\nTime permitting\, we will start
  the discussion of comparison of projections in \n$C^*$-algebras.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/143/
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