From Orbit spaces to Noncommutative Geometry

Noncommutative geometry replaces spaces by algebras in order
to study geometric situations where classical spaces fail to behave
well. A typical example arises from orbit spaces of group actions: while
proper actions produce nice quotients, many natural examples lead to
non-Hausdorff or pathological spaces.
In this talk, we explain how such spaces can be replaced by a C*-algebra
encoding the underlying dynamical system. We then show how geometric
information, such as topology and measure, can be recovered from these
algebras using tools like K-theory and traces.