The field of operator algebras was begun in the late 1920’s by J. von
Neumann, through his work on abstract Hilbert spaces and the
foundation of quantum mechanics. The subject developed steadily for
nearly a century, and found deep connections with many branches of
mathematics. The talk will discuss, using concrete examples, some of
these connections, in particular with group theory and geometry of
nonpositive curvature. The main focus will be the recent work of
Sylvain Barré and myself on intermediate rank geometry.
I will give some motivation for this work, and explain some of the
leading ideas in the study of discrete groups of intermediate rank.