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Index theory of differential operators in noncommutative geometry

Fries, Magnus LU orcid (2025)
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 K-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... (More)
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 K-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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Assoc. Prof. Kaad, Jens, University of Southern Denmark, Denmark.
organization
publishing date
type
Thesis
publication status
published
subject
keywords
index theory, boundary value problems, hypoellipticity, Kasparov theory, noncommutative geometry, spectral triples, spectral theory
pages
274 pages
publisher
Centre of Mathematical Sciences
defense location
Lecture Hall MH:Hörmander, Centre of Mathematical Sciences, Märkesbacken 4, Faculty of Engineering LTH, Lund University, Lund.
defense date
2026-01-30 13:00:00
ISBN
978-91-8104-762-2
978-91-8104-761-5
language
English
LU publication?
yes
id
359927cd-ee37-4dcb-9fdd-7b6aa7cebea4
date added to LUP
2025-12-15 17:29:39
date last changed
2025-12-29 11:54:51
@phdthesis{359927cd-ee37-4dcb-9fdd-7b6aa7cebea4,
  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 K-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.}},
  author       = {{Fries, Magnus}},
  isbn         = {{978-91-8104-762-2}},
  keywords     = {{index theory; boundary value problems; hypoellipticity; Kasparov theory; noncommutative geometry; spectral triples; spectral theory}},
  language     = {{eng}},
  publisher    = {{Centre of Mathematical Sciences}},
  school       = {{Lund University}},
  title        = {{Index theory of differential operators in noncommutative geometry}},
  url          = {{https://lup.lub.lu.se/search/files/235939632/MagnusFriesThesisLUCRIS.pdf}},
  year         = {{2025}},
}