The theory of rings of bounded operators on Hilbert space was
initiated by Murray and von Neumann in the 1930s, and has since
developed in myriad directions. Among these we single out the
theory of noncommutative topological spaces (C*-algebras) and
noncommutative measure spaces (von Neumann algebras), and examine
the structure of those objects which can reasonably be called
amenable. Here we find a common theme: deep information about
the structure of these algebras is often rooted in their tensorial
absroption of a canonical tractable algebra. For II_1 factors
this object is the hyperfinite II_1 factor, and for purely infinite
nuclear C*-algebras it is the Cuntz algebra O_\infty. For general
nuclear simple separable C*-algebras, the correct object is the
Jiang-Su algebra Z. In this talk we will discuss progress on proving
tensorial absorption of Z, and its consequences for the classification
theory of C*-algebras.

The **Back2Fields Colloquium Series** celebrates the accomplishments
of former postdoctoral fellows of Fields Institute thematic programs.
Over the past two decades, these programs attracted the rising
stars of their field and often launches very distinguished research
careers. As part of the 20th anniversary celebrations, this series
of colloquium talks will allow the general mathematical public
to become familiar with some of their work.

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