Public defense of PhD-thesis, Magnus Fries

Title: Index theory of differential operators in noncommutative geometry

Faculty opponent: Jens Kaad, University of Southern Denmark

Abstract: This thesis explores index theory for linear differential operators using tools from noncommutative geometry. We study how spectral triples can accommodate elliptic and Heisenberg-elliptic higher-order differential operators in 𝛫-homology, with a specific focus on manifolds with boundary. In the case of higher-order elliptic differential operators on manifolds with smooth compact boundary, we prove a generalization of the Baum-Douglas-Taylor index formula. From this, we obtain an obstruction to existence of elliptic boundary conditions. On non-compact manifolds, we revisit Gromov-Lawson’s relative index theorem and show that it holds in a more general setting. In connection to this, we obtain a geometric characterization of Fredholm operators. For anisotropic geometries, we study how spectral triples can be constructed from multiple operators of different orders that together capture the geometry. We also show that any elliptic or Heisenberg-elliptic differential operator can locally reconstruct the geodesic or the Carnot-Carathéodory distance, respectively. Lastly, we present a novel approach for an eigenvalue inequality for different boundary conditions of the Laplacian.