Non-commutative Geometry, Index Theory and Mathematical Physics

Abstract

Noncommutative geometry today is a new but mature branch of mathematics shedding light on many other areas from number theory to operator algebras. In the 2018 meeting two of these connections were high-lighted. For once, the applications to mathematical physics, in particular quantum field theory. Indeed, it was quantum theory which told us first that the world on small scales inherently is noncommutative. The second connection was to index theory with its applications in differential geometry. Here, non-commutative geometry provides the fine tools ot obtain higher information.