Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: A seminal result by Georges de Rham in 1931 was to establish an isomorphism between the singular cohomology and the de Rham cohomology on compact boundaryless manifolds. Given a smooth metric on such a manifold, a related result asserts that the space of harmonic forms is isomorphic to this cohomology. Beyond the benefit of introducing operator methods to the study of topology, this can be interpreted as prescribing a geometry to calculate the cohomology, making it possible to potentially choose an optimal metric for this task. However, there is no reason that such a metric needs to be smooth. Motivated by this, along with other considerations, we revisit Hodge theory in the context of singular metrics called Rough Riemannian metrics. Utilising the seminal perspective developed by Keith-McIntosh-Rosén, a perturbative approach is taken to show that the kernels of the induced Hodge-Dirac operators remain isomorphic under uniform changes of metric. This has the added benefit of allowing us to consider non-compact manifolds as well as manifolds with boundary. These results are obtained as a special case of a more general context featuring nilpotent differential operators supported on a vector bundle equipped with a Rough Hermitian metric.