Dimitris Gerontogiannis (IM PAN)

Title: The log-Laplacian on Ahlfors regular spaces and noncommutative boundaries

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this operator is intrinsically defined, its spectral properties are analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden). If time allows, I will also discuss the recent joint work with Magnus Goffeng (Lund) and Bram Mesland on applying the log-Laplacian to study the spectral geometry of Cuntz-Krieger algebras. The latter are C*-algebras associated with stable/unstable foliations on topological Markov chains.