###Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections
Project ID: 2228bd1112 (You will need this ID for your application)
Research Theme: Mathematical Sciences
UCL Lead department: Mathematics
Lead Supervisor: Dmitri Vassiliev
Project Summary:
Consider a single photon living in curved space. It is described by Maxwell’s equations. We seek solutions harmonic in time. This reduces to the spectral problem for the operator curl, whose spectrum is, in general, asymmetric about zero (think particle/antiparticle).
Spectral asymmetry is a classical subject in analysis and geometry, whose origins lie in the papers of Atiyah, Patodi and Singer. The project will provide a new approach to the study of spectral asymmetry based on the use of pseudodifferential techniques developed in a series of recent joint papers by Matteo Capoferri and myself.
M. Capoferri and D. Vassiliev. Invariant subspaces of elliptic systems I: pseudodifferential projections. Journal of Functional Analysis, 2022, vol. 282, article number 109402 (43 pages). Available as preprint arXiv:2103.14325.
M. Capoferri and D. Vassiliev. Invariant subspaces of elliptic systems II: spectral theory. Journal of Spectral Theory, 2022, vol. 12, p. 301-338. Available as preprint arXiv:2103.14334.