Listening to Operators: Multiple Operator Integrals and Noncommutative Geometry

Abstract: Many geometrical objects are equally interesting from the perspective of functional analysis. The Laplace operator for instance, though motivated geometrically, is sometimes better viewed as a specific self-adjoint unbounded operator. Even the notion of (a compact, Hausdorff) space itself is equivalent to the notion of a commutative unital C*-algebra, i.e., a specific operator algebra. This duality between geometric objects and operators is a rich source of research topics. Somewhere at the heart of this duality lies the topic of Multiple Operator Integrals. Its applications are often geometrical, while its tools are functional analytical. At the furthest outpost of the mentioned duality we find Noncommutative Geometry. We will explore both topics (MOI and NCG), and see what comes out if we combine them. Physical applications will be amply addressed.