Masaki Tsukamoto

 Hilbert cube is $[0,1]^Z$; the infinite product of the
interval $[0,1]$.
Question: Given a dynamical system, can we embed it
into the shift action on the Hilbert cube?
This problem has been studied more than 40 years.
In the talk I will present a fairly complete answer for minimal
dynamical systems.

In 1999 Elon Lindenstrauss proved that minimal system of mean dimension
less than $1/36$ can be embedded into the Hilbert cube.
The value $1/36$ looked artificial. So he asked what is the optimal
value for the embedding.

Recently Yonatan Gutman and I solved this problem by proving that
minimal systems of mean dimension less than $1/2$ can be embedded into
the Hilbert cube.
The value $1/2$ is optimal.

The proof is very interesting.
The nature of the above problem is purely abstract topological dynamics.
But main ingredients of the proof are Fourier and complex analysis.
So our work exhibits a new interaction between topological dynamics
and classical analysis.