
Introduction
There are several notions of "topological dimension" for noncommutative
C*algebras, each with their own strengths. Recent work of Wilhelm Winter
and collaborators has resulted in a new "nuclear" dimension that is very
natural, generalizes classical covering dimension of spaces, and is
known to be finite (in fact, < 6) for all simple, nuclear C*algebras
that have been classified so far. This work is inspired by classification,
but has led to substantial insights into the structure of arbitrary nuclear
C*algebras (i.e., inspired by said work, Kirchberg has proved they enjoy
a stronger approximation property than was previously known) and a new
definition of "Rokhlin dimension" for homeomorphisms of compact spaces
which is purely topological and extends the classical Rokhlin property (which
is the zerodimensional case).
Given the history of applications of
operator algebras to dynamics, from ConnesFeldmanWeiss to Popa's
groundbreaking work on the W*side and GiordanoPutnamSkau and related
results on the C*side, the above breakthroughs are particularly exciting.
The lectures are designed to be very accessible to graduate students with
limited backgrounds and to mathematicians who know only the basics of
C*algebra theory. Open problems and new research opportunities for
beginning researchers and students will be discussed.
