SCV seminar 2018

This is the schedule of the Complex Analysis Seminar for 2017:
Seminar organizer: Séverine Biard


Sverrir Örn Þorvaldsson: Boundary Fibration Structures and Quasi-Homogeneous Geometries, IV

Thursday, 25. January, 11:00-12:00, meeting room, 2nd floor in Tæknigarður.


Sverrir Örn Þorvaldsson: Boundary Fibration Structures and Quasi-Homogeneous Geometries, III

Thursday, 18. January, 11:00-12:00, Tgv227 in Tæknigarður.


Sverrir Örn Þorvaldsson: Boundary Fibration Structures and Quasi-Homogeneous Geometries, II

Thursday, 11. January, 13:00-14:00, Tgv227 in Tæknigarður.


Sverrir Örn Þorvaldsson: Boundary Fibration Structures and Quasi-Homogeneous Geometries, I

Friday, 5. January, 11:15-12:15, Tgv227 in Tæknigarður.

Abstract: In this work we extend work by Mazzeo on conformally compact manifolds to a class of manifolds with quasi-homogeneous geometries, which we call \(\kappa\)-manifolds. Our results show that there are complete noncompact manifolds of negative curvature, that have 0 in the essential spectrum for the Hodge Laplacian on forms, and this applies in a range of degrees centered at the middle degree. As is typical for boundary fibration structures our methods give much more, namely we provide a general framework to study elliptic partial differential operators on $\kappa$-manifolds based on microlocal methods. We construct a calculus of pseudodifferential operators on the manifold, and give precise conditions for the existence of a parametrix for elliptic differential operators in this calculus. We then apply this to the spectral theory of the Hodge Laplacian on a \(\kappa\)-manifold. This step requires detailed analysis of the Hodge Laplacian on a simpler model space, which in turn requires detailed study of a system of ordinary differential equations.

In the first talk we discuss the background, context and essence of the work. This entails some review of elliptic theory for compact Riemannian manifolds, and relations between geometry, topology and analysis on such manifolds. We provide some extensions of these to non-compact manifolds, provide examples of boundary fibration structures and discuss how these provide a foundation to construct suitable algebras of pseudodifferential operators that can be used to obtain the elliptic theory needed to study geometric partial differential operators on non-compact spaces.

In subsequent talks we provide more details of such a construction for $\kappa$-manifolds, but end with a study of a system of differential equations that are relevant to the problem, but interesting in itself and self-contained.