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{\Large Department of Mathematics, BGU}

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{\Huge Colloquium}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, May 31, 2016}
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\textbf{At} \emph{14:30 -- 15:30}
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\textbf{In} \emph{Math -101}

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{\large\scshape N. Christopher Phillips 
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  ( University of Oregon)
}
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will talk about
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{\Large\bfseries The mean dimension of a homeomorphism and  the radius of comparison of its C*-algebra\par}
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\textsc{Abstract:}
We describe a striking conjectured relation
between ``dimensions'{}' in topological dynamics and C\emph{-algebras.
(No previous knowledge
of C}-algebras or dimension theory will be assumed.)
Let \$X\$ be a compact metric space,
and let \$h \textbackslash{}colon X \textbackslash{}to X\$ be a minimal homeomorphism
(no nontrivial invariant closed subsets).
The \emph{mean dimension}
\$\{\textbackslash{}mathit\{mdim\}\} (h)\$ of \$h\$ is a dynamical invariant,
which I will describe in the talk,
and which was invented
for purposes having nothing to do with C\emph{-algebras.
One can also form a crossed product C}-algebra \$C\^{}* (\{\textbackslash{}mathbb\{Z\}\}, X, h)\$.
It is simple and unital,
and there is an explicit description
in terms of operators on Hilbert space,
which I will give in the talk.
The \emph{radius of comparison}
\$\{\textbackslash{}mathit\{rc\}\} (A)\$ of a simple unital C\emph{-algebra \$A\$
is an invariant introduced for reasons having nothing to do
with dynamics;
I will give the motivation for its definition in the talk
(but not the definition itself).
It has been conjectured,
originally on very thin evidence,
that the radius of comparison of \$C\^{}}(\{\textbackslash{}mathbb\{Z\}\},X,h)\$ 
 is equal to half the 
mean dimension of \$h\$
for any minimal homeomorphism \$h\$.

In this talk,
I will give elementary introductions to mean dimension,
the crossed product construction,
and the ideas behind the radius of comparison.
I will then describe the motivation for the conjecture
and some partial results towards it.








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