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 zero-dimensional case).
Given the history of applications of operator algebras to dynamics, from Connes-Feldman-Weiss to Popa's groundbreaking work on the W*-side and Giordano-Putnam-Skau 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.