Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections

Dmitri Vassiliev (University College London)

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. In this talk I will discuss 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.